Editorial Type:
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Article Type:
Research Article

The Role of Momentum Transfer in Tropical Cyclogenesis: Insights from a Single-Column Model

Hao Fu Department of the Geophysical Sciences, University of Chicago, Chicago, Illinois
Department of Earth System Science, Stanford University, Stanford, California

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https://orcid.org/0000-0002-9524-0001
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Christopher A. Davis NSF National Center for Atmospheric Research, Boulder, Colorado

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Online Publication:
12 Mar 2025
Print Publication:
01 Mar 2025
DOI:
https://doi.org/10.1175/JAS-D-24-0069.1
Page(s):
657–674
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Abstract

A step not fully understood in tropical cyclogenesis is the development of a surface cyclone, which is often preceded by a midlevel cyclone. This paper presents a single-column model to study the role of the transfer of tangential momentum in generating an initial surface cyclone. To isolate momentum transfer factors from thermodynamic factors, diabatic heating is set to be steady. The investigation starts without considering surface friction. The momentum transfer is decomposed into the transport by the vortex-scale circulation and by convection. The convective momentum transport (cumulus friction), when parameterized as a vertical eddy diffusion, leads to a vertical spectral truncation that permits an analytical solution of the single-column model. The analytical solution shows that the production of barotropic vorticity by the vortex-scale circulation is crucial for surface cyclone formation, and cumulus friction plays a dual role. Cumulus friction can enhance the downward momentum transfer, but when the eddy diffusion is too strong, the vortex-scale circulation is too damped to produce a significant barotropic cyclone. Between these two extremes lies an optimal eddy diffusivity that maximizes the growth rate of the surface cyclone. Finally, we add surface friction to the single-column model. Using scale analysis, we identify a critical vortex Rossby number above which surface friction becomes nonnegligible and significantly damps the development of the surface cyclone.

© 2025 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Hao Fu, haofu736@gmail.com

Abstract

A step not fully understood in tropical cyclogenesis is the development of a surface cyclone, which is often preceded by a midlevel cyclone. This paper presents a single-column model to study the role of the transfer of tangential momentum in generating an initial surface cyclone. To isolate momentum transfer factors from thermodynamic factors, diabatic heating is set to be steady. The investigation starts without considering surface friction. The momentum transfer is decomposed into the transport by the vortex-scale circulation and by convection. The convective momentum transport (cumulus friction), when parameterized as a vertical eddy diffusion, leads to a vertical spectral truncation that permits an analytical solution of the single-column model. The analytical solution shows that the production of barotropic vorticity by the vortex-scale circulation is crucial for surface cyclone formation, and cumulus friction plays a dual role. Cumulus friction can enhance the downward momentum transfer, but when the eddy diffusion is too strong, the vortex-scale circulation is too damped to produce a significant barotropic cyclone. Between these two extremes lies an optimal eddy diffusivity that maximizes the growth rate of the surface cyclone. Finally, we add surface friction to the single-column model. Using scale analysis, we identify a critical vortex Rossby number above which surface friction becomes nonnegligible and significantly damps the development of the surface cyclone.

© 2025 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Hao Fu, haofu736@gmail.com
Keywords: Deep convection; Stratiform clouds; Tropical cyclones; Friction; Vorticity; Single column models

1. Introduction

The process of tropical cyclone formation (cyclogenesis) represents the transition of a precursor vortex, which can arise from a multitude of morphologies of organized moist convection, into a surface-based vortex capable of amplification. Unlike a mature tropical cyclone (TC) with a strong surface circulation, a TC precursor vortex is usually a growing vorticity dipole with a midlevel cyclone and an upper-level anticyclone (Chen and Frank 1993; Fritsch et al. 1994; Bister and Emanuel 1997; Montgomery et al. 2006; Houze et al. 2009; Raymond et al. 2014; Bell and Montgomery 2019). The lower-tropospheric convergence is suppressed by the melting and evaporative cooling of stratiform precipitation (Houze 1982). A critical process in most TC genesis is the development of a surface cyclone from a midlevel cyclone at around 5-km height (Houze et al. 2009). Once the surface cyclone forms, boundary layer processes become critical for intensification. This includes the frictionally driven import of angular momentum (Montgomery and Smith 2014) and the wind-induced surface heat exchange (WISHE; Ooyama 1969; Emanuel 1986), the latter appearing necessary for a vortex to achieve realistic intensification rates (Emanuel 2012; Zhang and Emanuel 2016).

There are two main perspectives on the formation of a surface cyclone. Their relative importance remains unclear. The first perspective emphasizes the evolution of the diabatic heating profile, which we temporarily call the “thermodynamic” development (Raymond and Sessions 2007; Raymond et al. 2011, 2014; Gjorgjievska and Raymond 2014). The thermal wind relation shows that a midlevel cyclone is accompanied by a cold core at the lower level and a warm core at the upper level. Their influence on the static stability makes the diabatic heating profile more bottom heavy, reducing the convergence height and generating a low-level cyclonic circulation (Raymond et al. 2011).

This paper focuses on the second perspective: the vertical and horizontal transfer of tangential momentum, which might also result in the initial formation of a surface cyclone. We call the vortex evolution driven by momentum transfer, without invoking any thermodynamic feedback that changes the shape of the diabatic heating profile, a “mechanical” development process. Bister and Emanuel (1997) proposed that the evaporation-driven downdraft not only moistens the lower atmosphere but also advects the midlevel cyclone down to the surface. However, there is also evidence that momentum transfer disfavors surface cyclone development. Tory et al. (2006a) used budget analysis of a simulated TC precursor vortex to show that the deep convective updraft advects the midlevel cyclone upward rather than downward. Murthy and Boos (2019) showed that the low-level divergent downdraft driven by stratiform precipitation dilutes the deep-convection-produced surface cyclone. Murthy and Boos (2019) summarized these two mechanisms as the interaction of deep and stratiform heating, which disfavor surface cyclone growth.

The present article proposes a framework to reconcile the existing perspectives on the mechanical development of a surface cyclone. A hierarchy of models has been used to simulate TC genesis, including 3D and axisymmetric models (e.g., Ooyama 1969; Thorpe 1985; Rotunno and Emanuel 1987; Schubert et al. 1987; Montgomery et al. 2006). A further simplification is a 1D single-column model along the vortex center, highlighting the vertical development process. This approach has been widely used in modeling a cumulus cloud (e.g., Ogura and Takahashi 1971; Ferrier and Houze 1989; Fu and Lin 2019). As far as we know, the only single-column model for studying TC genesis was derived by Murthy and Boos (2019). We aim to move one step further by deriving an analytical solution of the single-column model to reconcile the puzzles in the mechanical development of a surface cyclone. Meanwhile, we realize that more insights can be gained by decomposing the momentum transfer into the transfer by vortex-scale circulation and by convective clouds.

The vertical transfer of horizontal momentum by clouds has been referred to as “cumulus friction” (e.g., Schneider and Lindzen 1976). Convection can vertically transfer air from one height to the other. Meanwhile, the interaction of the convective motion and vertical shear induces a pressure perturbation at the cloud scale, which influences the momentum of the in-cloud air. The cloud then influences the momentum of the background flow by detraining the in-cloud air (Wu and Yanai 1994; Han and Pan 2006). For isolated (unorganized) convective clouds, the convective momentum transfer is generally downgradient; for an organized storm, such as a squall line, both upgradient and downgradient transfers exist (LeMone et al. 1984; Moncrieff 1992). Romps (2014) used a bulk plume model to show that the convective momentum transfer of unorganized convection can be approximated as a vertically downgradient eddy diffusion. The role of convective momentum transfer in TC genesis still needs more investigation (Raymond et al. 2014). As far as we know, Lee (1984) is the only observational study that calculated the convective momentum flux of TC precursor vortices. Cumulus friction was shown to be a downgradient transport process, which helps spin up an initial surface cyclone.1 The present paper focuses on two questions pertaining to the initial formation of a surface cyclone:

  • What are the roles of cumulus friction and the momentum transfer by the vortex-scale circulation in TC genesis?

  • How large is cumulus friction compared to surface friction, and when does the latter become more important?

This paper is organized in the following way. In section 2, we derive a single-column model of a TC precursor vortex with both cumulus and surface friction. The convective momentum transfer is parameterized as a downgradient eddy diffusion. In section 3, we temporarily neglect surface friction and obtain an analytical solution, showing a competition between vortex-scale momentum transfer and cumulus friction. The competition yields an optimal eddy diffusivity that maximizes the growth rate of the surface vorticity. In section 4, we incorporate surface friction into the model, showing that it outweighs cumulus friction after the vortex Rossby number reaches a critical level. Section 5 concludes the article.

2. The single-column model

In this section, we derive a single-column model of TC genesis in a different way from Murthy and Boos (2019). They horizontally averaged the vertical vorticity equation in a wide cylinder, which covers the whole updraft region. Instead, we consider a narrow cylinder with a fixed radius that covers a portion of the vortex’s inner core. We believe the dynamics in the narrow cylinder could represent some crucial aspects of the vortex evolution.2 Under our narrow cylinder assumption, the main difference is the appearance of the vertical advection of tangential momentum by the vortex-scale circulation and convection. These factors will be shown to be crucial for understanding vortex evolution.

a. Derivation

We start from the flux-form vorticity equation (Haynes and McIntyre 1987). Because we focus on the middle- and lower-level dynamics, the Boussinesq approximation in z coordinate is used:
∂ζz∂t=−∇h⋅[uh(f+ζz)]︸horiz.advectiveflux+∇h⋅(ζhw)︸tiltingflux+∇h⋅(k×F)︸frictionalflux,
where ζz is the vertical relative vorticity (s−1), ζh (s−1) is the horizontal vorticity vector, f is the planetary vorticity (s−1), uh is the horizontal velocity vector (m s−1), w is the vertical velocity (m s−1), F is the turbulent friction force (m s−2), ∇h≡i(∂/∂x)+j(∂/∂y) is the horizontal gradient operator, and k is the vertical unit vector. The variation of f with latitude is ignored. The first term on the right-hand side of Eq. (1) denotes the horizontal advective flux; the second term denotes the tilting flux; and the third term denotes the frictional flux.
We study an erect vortex growing without a background flow. The “erect” assumption means the center of the vortex-scale vertical vorticity at each height should form a vertical line. We let the single-column model depict the horizontally averaged quantities in a narrow cylinder of radius r, which contains the plumb line crossing the vortex center at each height. See Fig. 1 for an illustration of the cylinder. The area average within the cylinder and the loop average along the periphery of the cylinder are defined as
⟨ζz⟩s≡1πr2∬ζzdS,⟨ζz⟩l≡12πr∮ζzdl.
For a narrow cylinder, the ζz and w within it are relatively uniform, so their area average and loop average are approximately equal:
⟨ζz⟩l≈⟨ζz⟩s,⟨w⟩l≈⟨w⟩s.
The perturbation quantities with respect to the loop average, i.e., the asymmetric quantities, are defined with a superscript of †:
ζz†≡ζz−⟨ζz⟩l,w†≡w−⟨w⟩l.
Performing the area average operation 〈〉s on both sides of Eq. (1), we obtain the area-averaged vorticity equation.
Fig. 1.
Fig. 1.

A schematic of the single-column model that depicts the horizontally averaged quantities in a narrow cylinder that covers a portion of the vortex’s inner core. The boundary layer top is defined as the origin of the vertical coordinate, with z = −HB depicting the sea surface and z = H depicting the tropopause. The blue-shaded region denotes the boundary layer, where the surface friction induces an Ekman circulation (black arrows) that decays with height. The red arrows denote cyclonic convergent flows, and the blue arrows denote anticyclonic divergent flows.

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0069.1

For the tendency term, we have
⟨∂ζz∂t⟩s=∂⟨ζz⟩s∂t.
For the horizontal advective flux term, we introduce the radial velocity ur:
−1πr2∬∇h⋅[uh(f+ζz)]dS=−1πr2∮ur(f+ζz)dl=−1πr2∮⟨ur⟩l(f+⟨ζz⟩l)dl−1πr2∮ur†ζz†dl≈(f+⟨ζz⟩s)∂⟨w⟩s∂z−1πr2∮ur†ζz†dl.
In deriving the last line of Eq. (6), we have used the mass continuity relation and Gauss’s theorem:
∂⟨w⟩s∂z=−1πr2∮⟨ur⟩ldl,
and used the assumption of a sufficiently narrow cylinder (〈ζz〉s ≈ 〈ζz〉l). From the momentum perspective, the horizontal advective flux comes from the radial advection of tangential momentum, as well as the Coriolis force that diverts the radial momentum to the tangential direction.
For the tilting flux term, we introduce the tangential velocity uθ:
1πr2∬∇h⋅(ζhw)dS=−1πr2∬k⋅∇×(w∂uh∂z)dS=−1πr2∮w∂uθ∂zdl=−1πr2∮⟨w⟩l⟨∂uθ∂z⟩ldl−1πr2∮w†∂uθ†∂zdl≈−⟨w⟩s∂⟨ζz⟩s∂z−1πr2∮w†∂uθ†∂zdl,
where ζh ≡ k × ∂uh/∂z and ∇≡i(∂/∂x)+j(∂/∂y)+k(∂/∂z). In deriving the last line of Eq. (8), we have used Stokes’ theorem:
⟨ζz⟩s=1πr2∮⟨uθ⟩ldl,
and used the assumption of a sufficiently narrow cylinder (〈w〉s ≈ 〈w〉l). From the momentum perspective, the tilting flux comes from the vertical advection of tangential momentum.
The turbulent friction term is expressed as
1πr2∬∇h⋅(k×F)dS=1πr2∮Fθdl,
where Fθ is the tangential component of turbulent friction.
Equations (5), (6), (8), and (10) yield the governing equation of 〈ζz〉s:
∂⟨ζz⟩s∂t=(f+⟨ζz⟩s)∂⟨w⟩s∂z︸stretching−⟨w⟩s∂⟨ζz⟩s∂z︸verticaladv−1πr2∮ur†ζz†dl︸eddyhorizontaltransport−1πr2∮w†∂uθ†∂zdl︸eddyverticaltransport+1πr2∮Fθdl︸turbulentfriction.
The first two terms on the right-hand side are the vortex-scale stretching term and vertical advection term, with the understanding that these terms do not operate independently. The asymmetric effects appear as the third and fourth terms: The third term represents the eddy horizontal transport of tangential momentum, and the fourth term represents the eddy vertical transport of tangential momentum. The fifth term is the turbulent friction term. The cylinder spans a region small enough to simplify the mean components of the horizontal advective flux and the tilting flux but large enough to make the average over eddies nonfluctuating.
The eddy horizontal transport links the vortex column region to the surroundings. Outside the vortex column, the tilting and stretching effects produce an anticyclonic shield around the barotropic cyclonic core (Tory et al. 2006a). Two asymmetric factors could cause the interaction of the vortex column with its surroundings. First, Carton (1992) showed that the anticyclonic shell of a vortex is susceptible to shear instability, causing mixing that transports negative vorticity to the vortex column. Second, the transient updrafts and downdrafts are spatially local structures that produce vorticity dipoles by tilting the vertical shear of the vortex circulation (Montgomery et al. 2006; Tory and Frank 2010; Kilroy et al. 2014). The cyclonic part of the vorticity dipoles may drift to the vortex core, and the anticyclonic part may be expelled (McWilliams and Flierl 1979; Montgomery and Enagonio 1998; Tory and Montgomery 2008; O’Neill et al. 2016; Fu and O’Neill 2021). These two factors might be parameterized as a horizontal eddy diffusivity that could be either upgradient or downgradient. To maintain mathematical tractability, we neglect the eddy horizontal transport term and leave this potentially important term for future work:
−1πr2∮ur†ζz†dl≈0.
The eddy vertical transport term is assumed to be dominated by deep convection. The budget analysis of Lee (1984) showed that the convective vertical transport of tangential momentum is downgradient in their observed TCs. Romps (2014) used a bulk plume model to show that for a wind structure with a vertical length scale above the kilometer scale, the cumulus friction can be approximated as eddy diffusion. We adopt this parameterization and express it in the vorticity formulation:
−1πr2∮w†∂uθ†∂zdl≈1πr2∮D∂2⟨uθ⟩l∂z2dl=D∂2⟨ζz⟩s∂z2,
where D is the eddy diffusivity. Romps (2014) theoretically predicted D to be around 10–30 m2 s−1 and benchmarked the result with large-eddy simulations of a nonrotating radiative-convective equilibrium (RCE) setup.
The turbulent friction term represents the boundary layer turbulence and the clear air turbulence in the stratified free troposphere. It can also be parameterized as a vertical eddy diffusivity. Clayson and Kantha (2008) used dropsonde data to estimate the magnitude of clear air turbulence to be around 1–10 m2 s−1. Romps (2014) pointed out that the vertical mixing effect of deep convection is much stronger than clear-air turbulence, so we ignore turbulent friction in the free troposphere. In the boundary layer, we use the bulk aerodynamic formula to parameterize the surface turbulent friction:
1πr2∮Fθdl≈{0,free troposphere,−CDRm|ζz,surf|ζz,surfHB,boundary layer,
where CD ≈ 10−3 is the surface drag coefficient (Fairall et al. 2003), HB ≈ 0.5–1 km is the boundary layer depth (Smith and Vogl 2008), Rm is the radius of maximum wind that is prescribed as a constant, and ζz,surf is the surface vorticity. We have used Rm|ζz,surf| to estimate the characteristic surface wind speed, assuming that the radius of the narrow cylinder is in the same order of magnitude as Rm.
The piecewise nature of Eq. (14) inspires us to make the single-column model two parts: the free-tropospheric part with a z dependence and a quasi-steady slab boundary layer part. The quasi-steady slab boundary layer approach has long been used in axisymmetric TC models (e.g., Schubert and Hack 1983; Kilroy and Smith 2017) but seems to have not been used in any single-column model. We should be aware that the neglect of vorticity tendency in the boundary layer dynamics is less self-consistent when HB is larger. Substituting Eqs. (12)–(14) into Eq. (11), we obtain the single-column model:
∂ζz∂t≈−w∂ζz∂z+(f+ζz)∂w∂z+D∂2ζz∂z2︷cumulus friction,free troposphere,0≈(f+ζz,surf)wBHB−CDRm|ζz,surf|ζz,surfHB︸surfacefriction,boundary layer,
where 〈ζz〉s and 〈w〉s are simply expressed as ζz and w. The symbols 〈〉s, 〈〉l, and † will not be used anymore. In the boundary layer equation, wB denotes the vertical velocity at the boundary layer top. Kepert and Wang (2001) studied a tropical storm and showed that the vertical advection of tangential momentum in the boundary layer is (marginally) smaller than the horizontal advection. We interpret it as the fractional change in vertical velocity within the mixed layer being larger than the horizontal velocity. Thus, the vertical advection is neglected for simplicity.
The slab boundary layer model provides a diagnostic relationship between wB and ζz,surf, essentially the Ekman pumping relation. Thus, we view the boundary layer model as the lower boundary condition for the free-tropospheric part. Letting z = 0 represent the boundary layer top (z = −HB as the sea surface), the bottom boundary condition is set as
∂ζz∂z=0,wB=CDRm|ζz,surf|ζz,surff+ζz,surf,atz=0.
Here, for simplicity, we set a zero-gradient vorticity boundary condition at the boundary layer top, which neglects the vertical transfer of tangential momentum across the boundary layer top. Such a simplification of Ekman layer dynamics into a stress-free horizontal velocity (∂ζz/∂z = 0) and nonzero vertical velocity boundary condition has also been used in a reduced model of turbulent rotating convection (Stellmach et al. 2014). We reiterate that the parameterization of boundary layer processes in this single-column model is relatively crude because the intent is to study when those processes become important, not to study those processes in detail.
The model top, which represents the tropopause, is set as a stress free and nonpenetrative boundary at z = H:
∂ζz∂z=0,w=0,atz=H.

b. Parameterization of vertical motion

We use a linearized “omega equation” (e.g., Holton 2004) to parameterize w as the superposition of the diabatic heating and surface frictional effects. Assuming ζz ≪ f, the linearized vorticity equation, thermal wind equation, and buoyancy equation are
(∂∂t−D∂2∂z2)ζz=f∂w∂z,
f∂ζz∂z=∇h2b,
(∂∂t−D∂2∂z2)b=−N2w+Q,
where b denotes the horizontal anomaly of buoyancy, N denotes the buoyancy frequency, and Q denotes the horizontal anomaly of diabatic heating source. Here, we have assumed that buoyancy is vertically diffusive, crudely representing the scale-dependent nature of longwave radiative damping (Fels 1982). For simplicity, we further assume the vertical diffusivity of ζz and b is identical. Combining Eqs. (18)–(20), we eliminate the tendency terms and the vertical diffusion terms and use Rm to simplify the horizontal Laplacian operator: ∇h2≈−Rm−2, finally obtaining the single-column version of the omega equation:
f2∂2w∂z2−N2Rm2w=−QRm2,w=wB=CDRm|ζz,surf|ζz,surff+ζz,surf≈CDRm|ζz,surf|ζz,surff,atz=0,w=0,atz=H.
Because the free-tropospheric omega equation is linear, we have also linearized the Ekman pumping relation [Eq. (16)] with the assumption of ζz ≪ f to make them compatible. Because Eq. (21) is a linear equation, the w induced by diabatic heating and surface friction can be superposed, as discussed below.
The diabatic heating profile in the tropical atmosphere generally consists of a deep convective mode and a stratiform mode (Mapes 2000; Liu and Moncrieff 2004; Houze 2004; Murthy and Boos 2019),3 which are represented by a half and full sinusoidal wave:
Q=ΔQ[sin⁡(πzH)︸deepconvective−AQsin⁡(2πzH)︸stratiform].
Here, ΔQ is a fixed diabatic heating magnitude and AQ is the top-heaviness parameter of the diabatic heating profile. The approximate solution of Eq. (21) is
w≈ΔWsin⁡(πzH)−AΔWsin⁡(2πzH)︸diabatic-drivencirculation,wQ+CDRmf|ζz,surf|ζz,surfexp⁡(−zLE)︸Ekmancirculation,wE.
The expression of w has two components: the diabatic-heating-driven circulation (diabatic circulation wQ) and the surface-friction-driven circulation (Ekman circulation wE). The parameter ΔW is the vertical velocity magnitude of the diabatic circulation, A (different from AQ) is the top-heaviness parameter of the diabatic circulation, and LE is the Rossby penetration depth of the Ekman circulation:
ΔW≡ΔQN2+π2f2Rm2H2,A≡AQN2+π2f2Rm2H2N2+4π2f2Rm2H2,LE≡fRmN.
The approximation in the solution of w [Eq. (23)] comes from the assumption of LE ≪ H. Using a typical f ∼ 5 × 10−5 s−1, Rm ∼ 102 km, N ∼ 0.005 s−1 (considering the water vapor saturation effect that makes it smaller than the typical 0.01 s−1 value), and H ∼ 10 km, we get LE/H ∼ 0.1, justifying this assumption.
With the w expression at hand [Eq. (23)], we approximate the vorticity dynamics in the free troposphere as
∂ζz∂t=−(wQ+wE)∂ζz∂z+(f+ζz)(∂wQ∂z+∂wE∂z)+D∂2ζz∂z2,∂ζz∂z=0,atz=0andz=H.
Here, the surface friction effect is represented by the Ekman circulation that squashes the vortex tube in the free troposphere. The surface vorticity ζz,surf is taken as ζz at z = 0, the boundary layer top height. Finally, we emphasize that wQ and wE are derived under the quasigeostrophic assumption and the equal vertical diffusivity assumption for ζz and b. The former requires ζz ≪ f, an assumption not fully satisfied in the simulations below but is probably still valid for gaining physical insights.

3. Steady diabatic heating without surface friction

This section studies the idealized case without surface friction and uses steady diabatic heating. In section 4, we will show that surface friction becomes more important as the surface cyclone grows, so there might be an early stage where surface friction is negligible compared to cumulus friction. Steady heating is the most commonly used setting in previous idealized studies of vortex evolution (e.g., Schubert et al. 1987; Murthy and Boos 2019). With steady heating and no surface friction, the system is linear. However, we follow the convention to refer to −w∂ζz/∂z and ζz∂w/∂z as nonlinear terms.

a. Two nondimensional parameters

The column model is controlled by five parameters: f, ΔW, H, D, and A. We can always rescale ζz with f, so f is a similarity parameter that does not change the system’s behavior. The parameters ΔW and H can be combined as an overturning time scale of τ*:
τ*≡(ΔWπH)−1.
The parameters D and H can be combined as a Rayleigh damping time scale of τd for the first baroclinic mode structure cos(πz/H):
τd≡D−1(πH)−2.
If we further normalize time t with τ*, the system only has two independent nondimensional parameters: τ*/τd and A.4
What determines τ*/τd? Using a bulk plume model, Romps (2014) parameterized the eddy diffusivity due to deep convection as
D≈ΔWε.
Here, ε is the cloud updraft fractional entrainment rate. Because the convective activity within a developing vortex is much stronger than in the environment, we have used ΔW to represent the convective volume flux, an input parameter of the theory of Romps (2014). The term ε is generally smaller for a wider cloud (Morton et al. 1956; de Rooy et al. 2013; Peters et al. 2020), diagnosed to be around 1 km−1 for unorganized deep convection by Romps (2014). According to the theory of Romps (2014), Eq. (28) works when the ε is not too small, so that a smaller ε enhances cumulus friction by enlarging the wind speed difference between the in-cloud and environmental air. Because the vertical wavenumbers of the cos(πz/H) and cos(2πz/H) vorticity structures π/H and 2π/H are much smaller than ε, this criterion is satisfied. Combining Eqs. (26)–(28), we get
τ*τd=πεH≈0.31,
where H = 10 km and ε = 1 km−1 are used. Equation (29) indicates that τ* and τd are intrinsically linked. Stronger diabatic heating is accompanied by stronger convective momentum transport. We will investigate a wide range of τ*/τd because ε is generally inversely proportional to the updraft radius, which has a wide spectrum (Arakawa and Schubert 1974).

What determines A? We follow Mapes (1993) to set A = 0.5 as the reference value, which corresponds to a top-heavy diabatic heating profile typical of the tropical atmosphere. Without vertical diffusion and nonlinear effects, A = 0.5 yields zero surface vorticity because ∂w/∂z = ∂wQ/∂z = 0 at the surface [Eq. (23)]. The A = 0.5 setting with nearly zero surface convergence may have a universal meaning. It might be close to a stable equilibrium point of the tropical atmosphere, where the gross moist stability is generally positive (Raymond et al. 2009).

To reveal the physics of mechanical development, we will view τ*/τd and A as free parameters and explore the parameter space spanned by them.

b. Numerical simulation result

First, we numerically solve the column model [Eqs. (23)–(25)] with the finite-difference method. The discretization scheme is introduced in the online supplemental material document. We will frequently use a vertical spectral decomposition view to analyze the simulation result, as illustrated in Fig. 2. The amplitude of the n = 0, 1, and 2 modes of vertical vorticity is defined as Z0, Z1, and Z2, which obey
ζz=Z0+Z1cos⁡(πzH)+Z2cos⁡(2πzH)+….
Here, Z0=ζz¯ is the barotropic vorticity, with the bar denoting vertical average. The barotropic vorticity will be shown to be crucial.
Fig. 2.
Fig. 2.

A schematic diagram of the vertical mode decomposition of vertical velocity and vertical vorticity, without considering surface friction. The first row shows the vertical velocity of the n = 0 mode (vertically uniform part), n = 1 mode [sin(πz/H)], n = 2 mode [sin(2πz/H)], and their sum: w = ΔW sin (π z/H) − AΔW sin (2πz/H), plotted with solid blue lines. The solid black lines denote the lower and upper boundaries. The dashed red line denotes the zero reference line. The second row shows the vertical vorticity. The Z0, Z1, and Z2 are the amplitude of the n = 0 mode, n = 1 mode, and n = 2 mode. Their magnitudes are sketched and not from any rigorous solution. A typical TC precursor vortex has Z0 > 0, Z1 > 0, and Z2 < 0.

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0069.1

We perform three experiments, with τ*/τd=0.25, 1, and 2, running for 2τ* time. All experiments use A = 0.5. Figure 3 shows the time evolution of the vorticity profile. For the τ*/τd=0.25 experiment, a midlevel cyclone is prominent at around z = 5 km. As τ*/τd increases, the maximum vorticity value at the simulation end time is smaller, and the maximum vorticity height is lower. Figure 4a shows the time evolution of surface vorticity in the three experiments. The growth rate of surface vorticity is not monotonic with respect to the vertical diffusivity. At t/τ*=2 time, the τ*/τd=1 experiment yields the largest surface vorticity. By running more experiments that traverse τ*/τd from near zero to two, we confirm an optimal τ*/τd≈0.7 that maximizes the surface vorticity for A = 0.5 (Fig. 5b). The diffusivity has dual roles in spinning up the surface cyclone:

Fig. 3.
Fig. 3.

The time evolution of the vertical structure of ζz/f for the three numerical experiments with the column model, all using A = 0.5. The increment of the contour line is 0.25. The zero contour line is thickened. (a) The τ*/τd=0.25 experiment. (b) The τ*/τd=1 experiment. (c) The τ*/τd=2 experiment.

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0069.1

Fig. 4.
Fig. 4.

(a) Comparing the surface vorticity (normalized by f) predicted by the column model (solid lines) and the analytical solution of the reduced model (dashed lines). The blue, red, and yellow lines denote τ*/τd=0.25, 1, and 2 experiments. All of them use A = 0.5. (b) As in (a), but the dashed lines are replaced by dotted lines, representing the column model’s surface vorticity output subtracting the contribution of the barotropic vorticity: (ζz,surf−ζz¯)/f.

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0069.1

Fig. 5.
Fig. 5.

The dependence of surface vorticity (normalized by f) on τ*/τd at t/τ*=2 time. (a) The A = 0.25 experiments. (b) The A = 0.5 experiments. (c) The A = 1 experiments. The solid lines denote the results of the column model, the dotted lines denote the results of the column model subtracting the barotropic vorticity [(ζz,surf−ζz¯)/f], and the dashed lines denote the analytical solution of the reduced model. The difference between the solid and dotted lines denotes the contribution from the barotropic vorticity.

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0069.1

  • Vertical diffusion suppresses the low-level anticyclone of the n = 2 mode more than the low-level cyclone of the n = 1 mode. This results in an apparent “diffusion” of the midlevel cyclone down to the surface.

  • Excessive diffusion suppresses all components of the baroclinic vorticity modes (n = 1 and n = 2). It slows down the nonlinear generation of the barotropic cyclone by the vortex-scale stretching and vertical advection, inhibiting the spinup of a surface cyclone.

Because the τ*/τd in the real atmosphere is estimated to be around unity [Eq. (29)], we infer that the nonmonotonic dependence of surface vorticity on τ*/τd should be considered in understanding TC genesis in the real atmosphere.

Why do the stretching and vertical advection of ζz produce a barotropic cyclone? We consider deep convective and stratiform heating separately and illustrate the mechanism in Fig. 6. For deep convective heating, the enhanced absolute vorticity at the lower level makes stretching more efficient and, therefore, the low-level cyclonic production more efficient. Similarly, the reduced absolute vorticity at the upper level makes squashing less efficient and, therefore, the upper-level anticyclonic production less efficient. The vertical advection moves the low-level cyclone upward, causing a midlevel cyclonic anomaly. For stratiform heating, the midlevel cyclone enhances the midlevel absolute vorticity, making the stretching (f + ζz)∂w/∂z more efficiently produce the midlevel cyclone. The vertical advection of the stratiform-heating-induced upper-level updraft and lower-level downdraft deepens the vertical range of the midlevel cyclone. For either the deep or stratiform heating, without considering their interaction, the net result is a barotropic cyclonic tendency. The cyclonic tendency does not change even if the vertical velocity profile is multiplied by a negative sign. Such invariance is due to a fundamental symmetry breaking in rotating convective fluids, which generally makes the vertical vorticity skew toward cyclonic (e.g., Julien et al. 1996; Vorobieff and Ecke 2002; Fu and Sun 2024).

Fig. 6.
Fig. 6.

A schematic diagram of (first row) how deep convective heating and (second row) stratiform heating produce barotropic vorticity via (first column) the stretching of vertical vorticity and (second column) the vertical advection of vertical vorticity. (third column) The sum of the two terms. Only the vorticity tendency produced by the same mode index is considered. The first row uses the n = 1 type of w and ζz; the second row uses the n = 2 type of w and ζz.

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0069.1

Is the barotropic vorticity crucial for generating a surface cyclone? We subtract the contribution of ζz¯ from the numerical output of ζz and find that the resulting surface vorticity is much smaller (Fig. 4b). Thus, barotropic vorticity is an indispensable component of the surface cyclone within this model. It is worth pointing out that research outside of TC genesis also reported the important role of barotropic vorticity. In idealized two-vertical-mode models of the large-scale tropical circulation, the nonlinear advection of the first-baroclinic-mode horizontal wind has been shown to generate a barotropic westerly wind (Neelin and Zeng 2000; Burns et al. 2006; Boos and Emanuel 2008). In the context of the Hadley cell, Boos and Emanuel (2008) further noticed the critical role of barotropic vorticity in strengthening the surface zonal wind, which could excite a WISHE feedback that intensifies the meridional flow.

Next, we perform experiments with varying A, the contribution of stratiform diabatic heating relative to deep convective heating (Fig. 5). Stronger stratiform heating involves stronger diabatic heating at the upper level due to freezing and stronger diabatic cooling at the lower level due to melting and rain evaporation. A counterintuitive result is that as A increases from 0.25 to 1, the maximum surface vorticity (among different τ*/τd at the simulation end time) first decreases and then increases. We know that stronger low-level diabatic cooling drives a stronger downdraft, making the surface wind more anticyclonic by squashing the planetary vorticity (f∂w/∂z < 0). How can a higher A enhance the surface cyclone? We subtract the contribution of barotropic vorticity from the model output of surface vorticity and find that the residual surface vorticity indeed drops as A increases (Fig. 5). Thus, the increase in surface vorticity with A must be due to the barotropic vorticity, which grows with A. Like diffusion, stratiform heating also plays dual roles in spinning up the surface cyclone:

  • A stronger stratiform heating linearly produces a stronger surface anticyclone by the low-level divergent motion that squashes the planetary vorticity (f∂w/∂z < 0).

  • A stronger stratiform heating nonlinearly produces a stronger surface cyclone by generating a stronger barotropic cyclone.

In summary, the numerical integration identifies the nonmonotonic dependence of surface vorticity on τ*/τd and A. Both the nonmonotonic relations involve the production of barotropic vorticity. Note that this result is derived under the assumption of steady diabatic heating and neglecting surface friction. Next, we derive a reduced equation to interpret the underlying physics.

c. The reduced equation set

In this subsection, we use the Fourier transform to derive a reduced equation set, which grasps the main feature of the column model.5 We let Zn be the amplitude of the nth vertical mode of vorticity:
Zn≡{1H∫0Hζzcos⁡(nπzH)dz=ζzcos⁡(nπzH)¯,n=0,2H∫0Hζzcos⁡(nπzH)dz=2ζzcos⁡(nπzH)¯,n≥1,ζz=∑n=0∞Zncos⁡(nπzH),
as illustrated in Fig. 2. Here, the overbar denotes the vertical average. Projecting the vorticity equation [Eq. (25)] onto the nth vertical mode, we get
2H∫0H∂ζz∂tcos⁡(nπzH)dz=−2H∫0Hw∂ζz∂zcos⁡(nπzH)dz+2H∫0H(f+ζz)∂w∂zcos⁡(nπzH)dz+2H∫0HD∂2ζz∂z2cos⁡(nπzH)dz.
Grouping each mode and using the orthogonal relation of trigonometric functions, we get a general expression of the mode equation:
(1+δn,0)dZndt=fτ*(δn,1−2Aδn,2)+1τ*(1+δn−1,0)(1−n2)Z|n−1|+1τ*(1+n2)Zn+1+Aτ*(1+δn−2,0)(2−n2)Z|n−2|−Aτ*(2+n2)Zn+2−n2τdZn,n=0,1,2,…,
where δn,p is the Kronecker symbol:
δn,p={1,n=p,0,n≠p.
The detailed derivation is documented in the supplemental material.

Equation (33) is an equation chain, with the evolution of Zn depending on the neighboring modes (n ± 1 and n ± 2). The damping from vertical diffusion increases with n as n2, so the chain can be truncated, essentially a Galerkin approximation. Figure 7 shows the time evolution of the n = 0, 1, 2, and 3 modes diagnosed from the τ*/τd=0.25, 1, and 2 experiments. The n = 3 mode is small yet finite in the τ*/τd=0.25 experiment and negligible in the other two experiments where the diffusion is stronger.

Fig. 7.
Fig. 7.

The time evolution of the vorticity amplitude Zn for n = 0 (blue line), n = 1 (red line), n = 2 (yellow line), and n = 3 (purple line) vertical modes. The solid lines denote the column model’s results, which are obtained from the Fourier transform of the solution. The dashed lines denote the reduced equation’s results. (a) The τ*/τd=0.25 experiment. (b) The τ*/τd=1 experiment. (c) The τ*/τd=2 experiment. All experiments use A = 0.5.

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0069.1

The n = 1 and 2 modes can be directly generated by diabatic heating via stretching the planetary vorticity. The n = 0 and n = 3 modes are generated indirectly via the stretching and vertical advection of ζz, so they are much weaker than the n = 1 and 2 modes at the early stage (Fig. 7). Why does the n = 3 mode have a much smaller amplitude than the n = 0 mode? This is because the vertical diffusion does not act on the n = 0 mode but significantly damps the n = 3 mode. Thus, the minimum model for studying vortex evolution should retain the n = 0, 1, and 2 modes. The truncated equation chain [Eq. (33)] leads to a reduced equation set:
dZ0dt≈1τ*(Z1−2AZ2),
dZ1dt≈−1τ*(f+Z0 −32AZ1+32Z2︸nonlinear damping)−Z1τd,
dZ2dt≈−1τ*2A(f+Z0)−4Z2τd.
The assumption of constant τ* and A makes the reduced equation a constant-coefficient system, allowing us to obtain an analytical solution.
To obtain the analytical solution, we first express Eqs. (35)–(37) in the matrix form:
dZdt=MZ+f,
where M and f are
M≡1τ*[01−2A1−32A−τ*τd32−2A0−4τ*τd⋅],f≡fτ*[01−2A].
The matrix comprising M’s right eigenvector is denoted as P. The analytical solution of Z is
Z=PΛP−1f,
where Λ carries the time-dependent information:
Λ≡[eλ1t−1λ1000eλ2t−1λ2000eλ3t−1λ3].
Here, λ1, λ2, and λ3 are the three eigenvalues of M, ranked by their real part in descending order. Technically speaking, they could be solved analytically as the roots of a cubic equation (e.g., Lebedev 1991). We calculate the eigenvalues numerically for convenience. The surface vorticity ζz,surf is
ζz,surf=Z0+Z1cos⁡(π0H)+Z2cos⁡(2π0H)=Z0+Z1+Z2=sZ=sPΛP−1f,
where we have used Eq. (40) and defined s ≡ [1, 1, 1] to project Z to the surface vorticity.

Figures 4, 5, and 7 show that the reduced equation is generally a good approximation to the column model. The error is larger for a smaller τ*/τd because a weaker vertical diffusivity suppresses the higher-n vertical modes less. Because τ*>0, there is Z1 > 0 and Z2 < 0 in general (see Fig. 2). We comment on three properties of the reduced equation:

  • First, the barotropic vorticity Z0 is nonlinearly produced by Z1 and Z2 due to the combined effect of stretching and vertical advection. Because Z1 > 0 and Z2 < 0, Eq. (35) shows Z0 is positive. This explains why the maximum surface vorticity is higher in the A = 1 case than in the A = 0.5 case (Fig. 5): The A = 1 case has a higher amplitude of the second-mode vorticity (higher −Z2), which nonlinearly generates more barotropic vorticity. The process identified by Bister and Emanuel (1997) can be viewed as a subset of the production of Z0 by Z2: The downward advection of the midlevel cyclone (n = 2 vorticity) by the low-level downdraft (n = 2 vertical motion) produces a surface cyclone.

  • Second, the −(3/2)AZ1+(3/2)Z2 term in the Z1 Eq. (36) has the opposite sign to the stretching part f + Z0, generating a nonlinear damping on Z1. It produces an upper-level cyclonic anomaly and a lower-level anticyclonic anomaly. This term includes two processes identified by Murthy and Boos (2019) that reduce the surface vorticity: 1) the upward advection of the n = 2 vorticity (midlevel cyclone) by the n = 1 mode’s updraft; 2) the downward advection of the n = 1 vorticity by the n = 2 mode’s low-level downdraft. The nonlinear damping is coupled with cumulus friction. When cumulus friction is stronger (τ*/τd increases), the (3/2)Z2 term is suppressed. Thus, cumulus friction may favor the formation of a surface cyclone not only through direct diffusion but also through suppressing the nonlinear damping effect.

  • Third, the f + Z0 term in the Z1 and Z2 equation indicates that the barotropic mode provides a cyclonic environment, which raises the background absolute vorticity from f to f + Z0. Such an enhancement of inertial stability makes the direct production of Z1 and Z2 by stretching more efficient (Hack and Schubert 1986).

Here, we summarize the three contributions to the surface cyclone: The first factor is positive, the second is negative, and the third is neutral. Though these factors have been identified in separate studies, the reduced model clarifies their relative contributions. Vertical diffusion (cumulus friction), which truncates the system to the reduced equation, makes the clean display of these terms possible. Given the complicity in the vortex mechanical development process, we argue that it cannot be simply attributed to “top-down” or “bottom-up.”

The analytical solution clearly shows the nonmonotonic dependence of ζz,surf on τ*/τd and A. Figure 8a shows the analytical solution of ζz,surf/f at t/τ*=2 time with varying τ*/τd and A. The nonmonotonic dependence is due to the existence of a saddle point at A ≈ 0.5 and τ*/τd≈0.7. The saddle point happens to be around the parameter regime of the tropical atmosphere (section 3a). The surface vorticity is the highest at the A ≪ 0.5 and A ≫ 0.5 regime and lowest at the τ*/τd≪1 and τ*/τd≫1 regime. The saddle point results from the dual role of cumulus friction, as well as the dual role of stratiform heating, in surface cyclogenesis. Subtracting the barotropic vorticity from the surface vorticity, the saddle point pattern disappears (Fig. 8c). The comparison confirms that barotropic vorticity is crucial for generating a surface cyclone, especially for A≳0.5, where the squashing of planetary vorticity at the lower level tends to produce a surface anticyclone.

Fig. 8.
Fig. 8.

(a) The analytical solution of ζz,surf/f at t/τ*=2 time with varying τ*/τd and A. The contour interval is 0.2. A higher τ*/τd means stronger diffusion (cumulus friction), and a higher A means a more top-heavy diabatic heating profile. The white crosses denote the three experiments with A = 0.5 and τ*/τd=0.25, 1, and 2. (b) As in (a), but letting λ2 = λ3 = 0 in Λ [Eq. (41)] to isolate the effect of the largest eigenvalue λ1. In (b), the regime where one of the eigenvalues is imaginary is denoted as the white region. This is a weakly diffusive regime where the truncation assumption of the reduced equation is invalid and does not deserve further investigation. (second row) As in (first row), but that they show the surface vorticity subtracting the barotropic vorticity, and the (ζz,surf−ζz¯)/f=0 contour line is bold.

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0069.1

Is the saddle point pattern robust if the simulation end time is extended? Because the fastest growing mode6 of the linear system determines its long-time asymptotic behavior, the question can be converted to whether the surface vorticity is mostly contributed by the fastest growing mode, i.e., the largest eigenvalue λ1. Figure 8b shows the saddle point pattern still exists when letting λ2 = λ3 = 0. Thus, the saddle point pattern should remain dominant at a later time.

4. The role of surface friction

The above calculation of surface vorticity does not consider surface friction, which grows together with the surface cyclone. Surface friction brings two new nondimensional parameters, the rescaled drag coefficient CD˜ and the rescaled Rossby penetration depth LE˜:
CD˜≡CDfRmΔW=πτ*NLEHCD,LE˜≡LEH=fRmNH.
The CD˜ is obtained by normalizing the boundary layer top vertical velocity wB [Eq. (21)] with ΔW and using f to represent the scale of ζz. In expressing CD˜ and LE˜, we have used the expression of LE ≡ fRm/N [Eq. (24)] and τ*≡(πΔW/H)−1 [Eq. (26)]. Using the Coriolis parameter f ∼ 5 × 10−5 s−1, surface drag coefficient CD ∼ 0.001, overturning time scale τ*∼5 days, buoyancy frequency N ∼ 0.005 s−1, radius of maximum wind Rm ∼ 100 km, and the tropospheric depth H ∼ 10 km, we get LE ∼ 1 km, LE˜∼0.1, and CD˜∼0.68.

We perform four experiments with the above parameters except making CD change between 0, 0.001, 0.002, and 0.003. They yield a fixed LE˜=0.1 and CD˜=0,0.68, 1.36, and 2.04. Other parameters use τ*/τd=0.5 and A = 0.5, a regime relevant to the tropical atmosphere as discussed in section 3a. Figure 9 shows that surface friction damps the surface cyclone, which appears as the squashing of the vortex tube in the lower troposphere by the Ekman circulation (e.g., Smith 2000). Comparing different experiments, we find that a higher CD makes the solution deviate from the no-surface-friction simulation earlier and at a smaller value of surface vorticity. Next, a semiquantitative approach will be used to derive a critical vorticity magnitude, above which the surface frictional effect becomes nonnegligible.

Fig. 9.
Fig. 9.

The result of numerical experiments with τ∗/τd=0.5, A = 0.5, and varying surface drag coefficients CD. (a) The time evolution of the surface vorticity, with the time normalized by τ∗ and the vorticity normalized by f. The experiments with CD = 0, 0.001, 0.002, and 0.003 are shown as solid lines whose color gradually turns lighter. (b) The surface vorticity versus the barotropic vorticity. Both quantities are normalized by f. (c) The barotropic vorticity (normalized by f) versus the square of the vortex Rossby number. (d) The surface vorticity (normalized by f) versus the square of the vortex Rossby number. The dashed black lines in (b)–(d) are 1:1 reference lines.

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0069.1

a. Estimating surface vorticity

The key for comparing the magnitude of surface friction is estimating the surface vorticity magnitude. Because we are only concerned about the critical vorticity magnitude for surface friction to become nonnegligible, it is tolerable to ignore surface friction in the estimation. Section 3 shows that for the A ∼ 0.5 and τ*/τd∼0.5 regime, close to the real atmosphere, the surface vorticity is significantly contributed by the nonlinearly generated barotropic vorticity. Thus, we may use barotropic vorticity to estimate the surface vorticity:
ζz,surf∼ζz¯.
Figure 9b shows Eq. (44) is a reasonable estimation for the four experiments in the ζz ≪ f regime.
How do we estimate the barotropic vorticity? We note that in the small-amplitude and weak vertical diffusion regime (ζz ≪ f and τ*≪τd), the reduced equation set [Eqs. (35)–(37)] further reduces to
dZ0dt≈1τ*(Z1−2AZ2),dZ1dt≈fτ*,dZ2dt≈−2Afτ*.
Multiplying f to the first line, multiplying Z1 to the second line, multiplying Z2 to the third line, and summing them up, we find a first integral of Eq. (45):
ddt(Z122+Z222−fZ0)=0⇒Z0f=1f2(Z122+Z222).
Here, we have used an initial condition of ζz = 0, which yields Z0 = Z1 = Z2 = 0. Equation (31) shows that ζz2¯ and ζz¯ obey
ζz2¯=Z02+Z122+Z222,ζz¯=Z0.
Assuming Z02≪Z12,Z22, which is valid for ζz ≪ f, we combine Eqs. (46) and (47) to get
ζz¯≈fRo2,Ro≡ζz2¯1/2f.
Here, Ro is the vortex Rossby number that represents the overall strength of the vortex. Figure 9c confirms that Eq. (48) is a good approximation for Ro ≪ 1, enabling us to express ζz¯ with Ro.
Combining Eqs. (44) and (48), we obtain an estimation of surface vorticity:
ζz,surf∼fRo2,
which agrees with the simulations in the ζz ≪ f regime (Fig. 9d).

b. A critical vortex Rossby number

With the estimation of surface vorticity [Eq. (49)] at hand, we derive the critical Ro above which surface friction plays a more important role than cumulus friction in controlling surface vorticity. The effect of surface friction on surface vorticity is realized by the Ekman circulation in the lower troposphere, which squashes the vortex tube. Using Eqs. (23), (24), and (49), we get
surface friction∼f∂wE∂z∼fCDRmfζz,surf2LE∼CDRmf2LERo4∼CDfNRo4.
The magnitude of cumulus friction is estimated as
cumulus friction∼D∂2ζz∂z2∼π2Dζz2¯1/2H2∼π2fDH2Ro.
Combining Eqs. (50) and (51), we get the ratio of surface to cumulus friction:
surface frictioncumulus friction∼CDNH2π2DRo3.
The Ro3 scaling indicates that the transition from a cumulus-friction-dominated regime to a surface-friction-dominated regime should be quite abrupt. The critical Ro for surface friction to exceed cumulus friction is readily derived by letting the friction ratio in Eq. (52) equal unity:
Critical Ro∼(π2DCDNH2)1/3∼(1CDNτd)1/3∝CD−1/3.
Equation (53) shows that the critical Ro is proportional to CD−1/3. A higher CD makes surface friction become dominant at an earlier stage, i.e., a less mature vortex.
Next, we benchmark the scaling theory [Eq. (53)] with three groups of experiments that use a fixed A = 0.5 and τ*/τd=0.25, 0.5, and 1. Each group has 10 experiments with CD ranging uniformly between 0.001 and 0.01. The chosen A and τ*/τd are in a regime where cumulus friction plays an important role in the vortex evolution and with parameter values relevant to the tropical atmosphere (section 3a). Thus, we hypothesize that the critical Ro represents the point where the surface vorticity begins to deviate from the prediction with the CD = 0 solution (Fig. 9a). We define the relative deviation as “dev”:
dev≡|ζz,surf−ζz,surf,CD=0|ζz,surf,CD=0,
where ζz,surf,CD=0 denotes the surface vorticity of the CD = 0 simulation. The time dev exceeds the 0.1 threshold is defined as the transition point, and the corresponding Ro is defined as the critical Ro. Because the magnitude of the diagnosed critical Ro is sensitive to the threshold, and the theory is derived semiquantitatively, we focus on the trend and then qualitatively compare the magnitude. Figure 10 confirms that the critical Ro in the simulations is roughly proportional to CD−1/3. Substituting Eq. (43) into Eq. (53), and using LE˜∼0.1 and CD˜∼1, we estimate a critical Ro of around Ro∼(πLE˜/CD˜)1/3∼0.7, a reasonable order of magnitude. These agreements support our semiquantitative theory.
Fig. 10.
Fig. 10.

The three dotted lines show the critical Ro diagnosed with the dev = 0.1 threshold in three groups of experiments with τ*/τd=0.25 (blue), τ*/τd=0.5 (red), and τ*/τd=1 (yellow). Each line shows 10 experiments with CD uniformly sampled between 0.001 and 0.01. The plotting uses the log–log coordinate. The solid black line marks the CD−1/3 reference slope (its magnitude does not have any specific meaning).

Citation: Journal of the Atmospheric Sciences 82, 3; 10.1175/JAS-D-24-0069.1

We emphasize a constraint that simplifies the derivation of the critical Ro: The surface vorticity is dominated by barotropic vorticity, as shown in the bold contours in the parameter space (Figs. 8c,d). The tropical atmosphere, where (A ∼ 0.5 and τd/τ*∼0.5), happens to roughly satisfy the condition. This section only makes a preliminary comparison between the cumulus and the surface friction, showing the validity of omitting surface friction when analyzing the early stage. A close study of their interaction is left for future work. We also remind the readers that the purely negative role of surface friction in surface cyclone development within this model results from the assumption of a fixed diabatic heating structure. In the real atmosphere, surface friction can also favor TC genesis by enhancing moisture convergence and organizing deep convection (Charney and Eliassen 1964; Kilroy et al. 2017). The surface friction’s dual role will be explored in an extended version of the single-column model that incorporates the response of diabatic heating to the frictional circulation.

5. Summary and discussion

This paper studies an important step in tropical cyclogenesis: the extension of the midlevel cyclone to the surface. This vertical extension has been explained from two perspectives. The first perspective is thermodynamic development, which denotes the bottom-heavy tendency of the diabatic heating profile that draws the convergent level lower. The second perspective is mechanical development, which denotes the transfer of horizontal momentum that spins up a surface cyclone. We focus on mechanical development, a nonlinear process that still lacks an analytically tractable model. The problem can be rephrased as follows: If the vortex column remained unsaturated and the diabatic heating profile remained top heavy, how much surface cyclonic vorticity would momentum transfer produce? Compared to the widely used axisymmetric framework, the single-column framework is an even simpler tool, suitable for understanding the vertical development of a vortex, but has received little attention. We try to improve the single-column framework introduced by Murthy and Boos (2019) by considering a missing factor: cumulus friction, which turns out to facilitate rather than complicate the analytical procedure.

We apply the theory of Romps (2014) to parameterize the cumulus friction of unorganized convection as vertical eddy diffusion. Given that the surface vorticity develops from a small value, we neglect surface friction first to focus on the role of cumulus friction and then study at which moment the surface friction becomes nonnegligible. The vertical eddy diffusion enables a vertical spectral truncation, which reduces the single-column model to a reduced equation. When the diabatic heating is steady, the reduced equation has an analytical solution. A key finding is that barotropic vorticity is an indispensable component of the surface cyclone in most cases. The analytical solution shows a nonmonotonic dependence of surface vorticity on vertical diffusivity D and the top-heaviness parameter of the diabatic-heating-driven circulation A. First, there is an optimal D that yields the fastest growth of the surface cyclone. Diffusion preferentially damps the n = 2 mode vorticity, directly suppressing its surface anticyclonic component and indirectly suppressing its nonlinear damping effect on the n = 1 vorticity. However, an overly strong diffusion damps the whole vortex-scale circulation, preventing it from nonlinearly generating barotropic vorticity by the stretching and vertical advection of vertical vorticity. Second, there is an A that yields the slowest growth of the surface cyclone. A higher A corresponds to a stronger stratiform heating. It generates a stronger low-level downdraft that squashes the planetary vorticity and produces a stronger low-level anticyclone. However, an overly strong stratiform heating generates greater barotropic vorticity through the vortex-scale circulation, boosting the surface cyclone. The dependence of surface vorticity on D and A appears as a saddle point pattern, and the current climatological value is coincidentally near the saddle point.

Then, we study the role of surface friction by incorporating the Ekman circulation into the single-column model. The surface frictional effect grows with the square of surface vorticity, serving as a high-order damping that becomes important at the later stage. Experiments with varying surface drag coefficients CD show that a higher CD makes the surface vorticity deviate from the CD = 0 experiment at an earlier time. This inspires us to derive the critical vortex Rossby number by which surface friction becomes nonnegligible. Around A = 0.5 and τd/τ*∼0.5 (relevant to the regime of the tropical atmosphere), we find that the surface vorticity is roughly proportional to the barotropic vorticity, and the barotropic vorticity is roughly proportional to the square of the vortex Rossby number. This enables us to derive a critical vortex Rossby number of Ro∝CD−1/3 for surface friction to significantly influence the vortex evolution, agreeing with single-column model simulations.

All findings presented in this paper are within the scope of a highly idealized single-column model. Future work along two paths might be considered. The first path is to validate the assumptions and predictions of this single-column model using observational data or convection-permitting simulations. A diagnostic study is needed to check whether the convective momentum transfer in TC genesis can be parameterized as a downgradient eddy diffusion (Lee 1984). The downgradient theory of Romps (2014), which works for unorganized convection, might break down as the TC intensifies and the convection gets more organized by the vortex-induced shear (e.g., Davis 2015). The second path is to improve the single-column model by adding more physical factors. One possible extension is to consider the change in density with height. Another extension is to incorporate the evolution of diabatic heating, i.e., the thermodynamic development. The diabatic heating profile should depend on the vertical structure of buoyancy perturbation and the humidity of the air column (e.g., Raymond and Sessions 2007). In particular, one can include some important feedbacks represented in WISHE by letting the surface vorticity control the moistening rate of the air column (e.g., Raymond et al. 2007). The ultimate goal is to understand how the mechanical and thermodynamic pathways cooperate to produce and intensify the surface cyclone.

1

Some 3D convection-permitting simulation research on TC genesis presented the contour plot of the eddy vertical transport of tangential momentum. However, whether it is downgradient has not been discussed, and it is hard to tell qualitatively from the figures. See Fig. 16d of Montgomery et al. (2006) and Fig. 7d of Tory et al. (2006b). On the other hand, recent 3D convection-permitting simulations of intensifying tropical storms and hurricanes by Persing et al. (2013) and Montgomery et al. (2020) have not found evidence of downgradient transport. Thus, further investigations are needed on the role of convective momentum transfer at different stages of TC genesis.

2

The structure away from the vortex center, though important, might be peeled off by the environmental straining flow and no longer influence the vortex dynamics (e.g., Carton 1992; Rozoff et al. 2006; Wang 2014).

3

For simplicity, we have ignored a potentially important cloud type: congestus convection. Its cloud top is around 5 km and produces low-level diabatic heating (e.g., Johnson et al. 1999; Wu 2003; Wang 2014).

4

This result is consistent with the Buckingham Pi theorem, which states that the number of nondimensional quantities in a problem depends on the number of independent physical quantities subtracting the number of physical dimensions (e.g., Evans 1972). Six physical quantities (τ*, τd, f, A, ζz, and w) and two dimensions (length and time) yield 6 − 2 = 4 nondimensional quantities τ*/τd, A, ζz/f, and w/ΔW. Of the four nondimensional quantities, τ*/τd and A are the two nondimensional parameters.

5

A related modeling framework is the two-vertical-mode models for studying the large-scale tropical circulation, including the Hadley cell (Neelin and Zeng 2000; Burns et al. 2006; Boos and Emanuel 2008). They retained the barotropic mode and the first baroclinic mode (with a temperature basis instead of a trigonometric function of z) and studied their nonlinear interaction. TC genesis is a similar yet different problem. The main difference is that the second baroclinic mode stratiform heating, which produces the midlevel cyclone, needs to be considered.

6

The “mode” here denotes one of the three temporal eigenmodes of the reduced equation, not vertical modes.

Acknowledgments.

This material is based upon work supported by the NSF National Center for Atmospheric Research, which is a major facility sponsored by the U.S. National Science Foundation under Cooperative Agreement 1852977. This work was initiated when Hao Fu visited NCAR, and some preliminary results were reported in his doctoral thesis. Hao Fu is now supported by the T. C. Chamberlin Postdoctoral Fellowship from the University of Chicago. The demonstrative experiment of a hurricane-like vortex by Noboru Nakamura provided valuable inspiration. We thank Richard Rotunno for an internal review of the previous version of the manuscript and helpful discussion. We thank William R. Boos for his suggestion of studying the role of surface friction semiquantitatively. We thank Shiwei Sun, Morgan E O’Neill, Bolei Yang, and Jian-Feng Gu for the helpful discussion. Finally, we thank Kevin Tory and three anonymous referees for carefully reviewing the manuscript, which significantly improved its scientific quality.

Data availability statement.

The MATLAB code for the single-column model and for generating all figures are documented at Zenodo at https://doi.org/10.5281/zenodo.14585148.

REFERENCES

  • Arakawa, A., and W. H. Schubert, 1974: Interaction of a cumulus cloud ensemble with the large-scale environment, Part I. J. Atmos. Sci., 31, 674–701, https://doi.org/10.1175/1520-0469(1974)031<0674:IOACCE>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Bell, M. M., and M. T. Montgomery, 2019: Mesoscale processes during the genesis of Hurricane Karl (2010). J. Atmos. Sci., 76, 2235–2255, https://doi.org/10.1175/JAS-D-18-0161.1.

    • Search Google Scholar
    • Export Citation

  • Bister, M., and K. A. Emanuel, 1997: The genesis of Hurricane Guillermo: TEXMEX analyses and a modeling study. Mon. Wea. Rev., 125, 2662–2682, https://doi.org/10.1175/1520-0493(1997)125<2662:TGOHGT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Boos, W. R., and K. A. Emanuel, 2008: Wind–evaporation feedback and abrupt seasonal transitions of weak, axisymmetric Hadley circulations. J. Atmos. Sci., 65, 2194–2214, https://doi.org/10.1175/2007JAS608.1.

    • Search Google Scholar
    • Export Citation

  • Burns, S. P., A. H. Sobel, and L. M. Polvani, 2006: Asymptotic solutions of the axisymmetric moist Hadley circulation in a model with two vertical modes. Theor. Comput. Fluid Dyn., 20, 443–467, https://doi.org/10.1007/s00162-006-0024-z.

    • Search Google Scholar
    • Export Citation

  • Carton, X. J., 1992: On the merger of shielded vortices. Europhys. Lett., 18, 697, https://doi.org/10.1209/0295-5075/18/8/006.

  • Charney, J. G., and A. Eliassen, 1964: On the growth of the hurricane depression. J. Atmos. Sci., 21, 68–75, https://doi.org/10.1175/1520-0469(1964)021<0068:OTGOTH>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Chen, S. S., and W. M. Frank, 1993: A numerical study of the genesis of extratropical convective mesovortices. Part I: Evolution and dynamics. J. Atmos. Sci., 50, 2401–2426, https://doi.org/10.1175/1520-0469(1993)050<2401:ANSOTG>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Clayson, C. A., and L. Kantha, 2008: On turbulence and mixing in the free atmosphere inferred from high-resolution soundings. J. Atmos. Oceanic Technol., 25, 833–852, https://doi.org/10.1175/2007JTECHA992.1.

    • Search Google Scholar
    • Export Citation

  • Davis, C. A., 2015: The formation of moist vortices and tropical cyclones in idealized simulations. J. Atmos. Sci., 72, 3499–3516, https://doi.org/10.1175/JAS-D-15-0027.1.

    • Search Google Scholar
    • Export Citation

  • de Rooy, W. C., and Coauthors, 2013: Entrainment and detrainment in cumulus convection: An overview. Quart. J. Roy. Meteor. Soc., 139 (670), 1–19, https://doi.org/10.1002/qj.1959.

    • Search Google Scholar
    • Export Citation

  • Emanuel, K., 2012: Self-stratification of tropical cyclone outflow. Part II: Implications for storm intensification. J. Atmos. Sci., 69, 988–996, https://doi.org/10.1175/JAS-D-11-0177.1.

    • Search Google Scholar
    • Export Citation

  • Emanuel, K. A., 1986: An air-sea interaction theory for tropical cyclones. Part I: Steady-state maintenance. J. Atmos. Sci., 43, 585–605, https://doi.org/10.1175/1520-0469(1986)043<0585:AASITF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Evans, J. H., 1972: Dimensional analysis and the Buckingham Pi theorem. Amer. J. Phys., 40, 1815–1822, https://doi.org/10.1119/1.1987069.

    • Search Google Scholar
    • Export Citation

  • Fairall, C. W., E. F. Bradley, J. Hare, A. A. Grachev, and J. B. Edson, 2003: Bulk parameterization of air–sea fluxes: Updates and verification for the COARE algorithm. J. Climate, 16, 571–591, https://doi.org/10.1175/1520-0442(2003)016<0571:BPOASF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Fels, S. B., 1982: A parameterization of scale-dependent radiative damping rates in the middle atmosphere. J. Atmos. Sci., 39, 1141–1152, https://doi.org/10.1175/1520-0469(1982)039<1141:APOSDR>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Ferrier, B. S., and R. A. Houze Jr., 1989: One-dimensional time-dependent modeling of GATE cumulonimbus convection. J. Atmos. Sci., 46, 330–352, https://doi.org/10.1175/1520-0469(1989)046<0330:ODTDMO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Fritsch, J. M., J. D. Murphy, and J. S. Kain, 1994: Warm core vortex amplification over land. J. Atmos. Sci., 51, 1780–1807, https://doi.org/10.1175/1520-0469(1994)051<1780:WCVAOL>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Fu, H., and Y. Lin, 2019: A kinematic model for understanding rain formation efficiency of a convective cell. J. Adv. Model. Earth Syst., 11, 4395–4422, https://doi.org/10.1029/2019MS001707.

    • Search Google Scholar
    • Export Citation

  • Fu, H., and M. O’Neill, 2021: The role of random vorticity stretching in tropical depression genesis. J. Atmos. Sci., 78, 4143–4168, https://doi.org/10.1175/JAS-D-21-0087.1.

    • Search Google Scholar
    • Export Citation

  • Fu, H., and S. Sun, 2024: Vorticity skewness of finite-amplitude rapidly rotating Rayleigh–Bénard convection. J. Fluid Mech., 997, A3, https://doi.org/10.1017/jfm.2024.571.

    • Search Google Scholar
    • Export Citation

  • Gjorgjievska, S., and D. J. Raymond, 2014: Interaction between dynamics and thermodynamics during tropical cyclogenesis. Atmos. Chem. Phys., 14, 3065–3082, https://doi.org/10.5194/acp-14-3065-2014.

    • Search Google Scholar
    • Export Citation

  • Hack, J. J., and W. H. Schubert, 1986: Nonlinear response of atmospheric vortices to heating by organized cumulus convection. J. Atmos. Sci., 43, 1559–1573, https://doi.org/10.1175/1520-0469(1986)043<1559:NROAVT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Han, J., and H.-L. Pan, 2006: Sensitivity of hurricane intensity forecast to convective momentum transport parameterization. Mon. Wea. Rev., 134, 664–674, https://doi.org/10.1175/MWR3090.1.

    • Search Google Scholar
    • Export Citation

  • Haynes, P. H., and M. E. McIntyre, 1987: On the evolution of vorticity and potential vorticity in the presence of diabatic heating and frictional or other forces. J. Atmos. Sci., 44, 828–841, https://doi.org/10.1175/1520-0469(1987)044<0828:OTEOVA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Holton, J. R., 2004: An Introduction to Dynamic Meteorology. International Geophysics Series, Vol. 88, Elsevier Science, 535 pp.

  • Houze, R. A., Jr., 1982: Cloud clusters and large-scale vertical motions in the tropics. J. Meteor. Soc. Japan, 60, 396–410, https://doi.org/10.2151/jmsj1965.60.1_396.

    • Search Google Scholar
    • Export Citation

  • Houze, R. A., Jr., 2004: Mesoscale convective systems. Rev. Geophys., 42, RG4003, https://doi.org/10.1029/2004RG000150.

  • Houze, R. A., Jr., W.-C. Lee, and M. M. Bell, 2009: Convective contribution to the genesis of Hurricane Ophelia (2005). Mon. Wea. Rev., 137, 2778–2800, https://doi.org/10.1175/2009MWR2727.1.

    • Search Google Scholar
    • Export Citation

  • Johnson, R. H., T. M. Rickenbach, S. A. Rutledge, P. E. Ciesielski, and W. H. Schubert, 1999: Trimodal characteristics of tropical convection. J. Climate, 12, 2397–2418, https://doi.org/10.1175/1520-0442(1999)012<2397:TCOTC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Julien, K., S. Legg, J. McWilliams, and J. Werne, 1996: Hard turbulence in rotating rayleigh-bénard convection. Phys. Rev. E, 53, R5557, https://doi.org/10.1103/PhysRevE.53.R5557.

    • Search Google Scholar
    • Export Citation

  • Kepert, J., and Y. Wang, 2001: The dynamics of boundary layer jets within the tropical cyclone core. Part II: Nonlinear enhancement. J. Atmos. Sci., 58, 2485–2501, https://doi.org/10.1175/1520-0469(2001)058<2485:TDOBLJ>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Kilroy, G., and R. K. Smith, 2017: The effects of initial vortex size on tropical cyclogenesis and intensification. Quart. J. Roy. Meteor. Soc., 143, 2832–2845, https://doi.org/10.1002/qj.3134.

    • Search Google Scholar
    • Export Citation

  • Kilroy, G., R. K. Smith, and U. Wissmeier, 2014: Tropical convection: The effects of ambient vertical and horizontal vorticity. Quart. J. Roy. Meteor. Soc., 140, 1756–1770, https://doi.org/10.1002/qj.2261.

    • Search Google Scholar
    • Export Citation

  • Kilroy, G., M. T. Montgomery, and R. K. Smith, 2017: The role of boundary-layer friction on tropical cyclogenesis and subsequent intensification. Quart. J. Roy. Meteor. Soc., 143, 2524–2536, https://doi.org/10.1002/qj.3104.

    • Search Google Scholar
    • Export Citation

  • Lebedev, V. I., 1991: On formulae for roots of cubic equation. Russ. J. Numer. Anal. Math. Modell., 6, 315–324, https://doi.org/10.1515/rnam.1991.6.4.315.

    • Search Google Scholar
    • Export Citation

  • Lee, C.-S., 1984: The bulk effects of cumulus momentum transports in tropical cyclones. J. Atmos. Sci., 41, 590–603, https://doi.org/10.1175/1520-0469(1984)041<0590:TBEOCM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • LeMone, M. A., G. M. Barnes, and E. J. Zipser, 1984: Momentum flux by lines of cumulonimbus over the tropical oceans. J. Atmos. Sci., 41, 1914–1932, https://doi.org/10.1175/1520-0469(1984)041<1914:MFBLOC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Liu, C., and M. W. Moncrieff, 2004: Effects of convectively generated gravity waves and rotation on the organization of convection. J. Atmos. Sci., 61, 2218–2227, https://doi.org/10.1175/1520-0469(2004)061<2218:EOCGGW>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Mapes, B. E., 1993: Gregarious tropical convection. J. Atmos. Sci., 50, 2026–2037, https://doi.org/10.1175/1520-0469(1993)050<2026:GTC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Mapes, B. E., 2000: Convective inhibition, subgrid-scale triggering energy, and stratiform instability in a toy tropical wave model. J. Atmos. Sci., 57, 1515–1535, https://doi.org/10.1175/1520-0469(2000)057<1515:CISSTE>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • McWilliams, J. C., and G. R. Flierl, 1979: On the evolution of isolated, nonlinear vortices. J. Phys. Oceanogr., 9, 1155–1182, https://doi.org/10.1175/1520-0485(1979)009<1155:OTEOIN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Moncrieff, M. W., 1992: Organized convective systems: Archetypal dynamical models, mass and momentum flux theory, and parametrization. Quart. J. Roy. Meteor. Soc., 118, 819–850, https://doi.org/10.1002/qj.49711850703.

    • Search Google Scholar
    • Export Citation

  • Montgomery, M. T., and J. Enagonio, 1998: Tropical cyclogenesis via convectively forced vortex Rossby waves in a three-dimensional quasigeostrophic model. J. Atmos. Sci., 55, 3176–3207, https://doi.org/10.1175/1520-0469(1998)055<3176:TCVCFV>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Montgomery, M. T., and R. K. Smith, 2014: Paradigms for tropical cyclone intensification. Aust. Meteor. Oceanogr. J., 64, 37–66.

  • Montgomery, M. T., M. E. Nicholls, T. A. Cram, and A. B. Saunders, 2006: A vortical hot tower route to tropical cyclogenesis. J. Atmos. Sci., 63, 355–386, https://doi.org/10.1175/JAS3604.1.

    • Search Google Scholar
    • Export Citation

  • Montgomery, M. T., G. Kilroy, R. K. Smith, and N. Črnivec, 2020: Contribution of mean and eddy momentum processes to tropical cyclone intensification. Quart. J. Roy. Meteor. Soc., 146, 3101–3117, https://doi.org/10.1002/qj.3837.

    • Search Google Scholar
    • Export Citation

  • Morton, B. R., G. I. Taylor, and J. S. Turner, 1956: Turbulent gravitational convection from maintained and instantaneous sources. Proc. Roy. Soc. London, 234A (1196), 1–23, https://doi.org/10.1098/rspa.1956.0011.

    • Search Google Scholar
    • Export Citation

  • Murthy, V. S., and W. R. Boos, 2019: Understanding the vertical structure of potential vorticity in tropical depressions. Quart. J. Roy. Meteor. Soc., 145, 1968–1991, https://doi.org/10.1002/qj.3539.

    • Search Google Scholar
    • Export Citation

  • Neelin, J. D., and N. Zeng, 2000: A quasi-equilibrium tropical circulation model—Formulation. J. Atmos. Sci., 57, 1741–1766, https://doi.org/10.1175/1520-0469(2000)057<1741:AQETCM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Ogura, Y., and T. Takahashi, 1971: Numerical simulation of the life cycle of a thunderstorm cell. Mon. Wea. Rev., 99, 895–911, https://doi.org/10.1175/1520-0493(1971)099<0895:NSOTLC>2.3.CO;2.

    • Search Google Scholar
    • Export Citation

  • O’Neill, M. E., K. A. Emanuel, and G. R. Flierl, 2016: Weak jets and strong cyclones: Shallow-water modeling of giant planet polar caps. J. Atmos. Sci., 73, 1841–1855, https://doi.org/10.1175/JAS-D-15-0314.1.

    • Search Google Scholar
    • Export Citation

  • Ooyama, K., 1969: Numerical simulation of the life cycle of tropical cyclones. J. Atmos. Sci., 26, 3–40, https://doi.org/10.1175/1520-0469(1969)026<0003:NSOTLC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Persing, J., M. T. Montgomery, J. C. McWilliams, and R. K. Smith, 2013: Asymmetric and axisymmetric dynamics of tropical cyclones. Atmos. Chem. Phys., 13, 12 299–12 341, https://doi.org/10.5194/acp-13-12299-2013.

    • Search Google Scholar
    • Export Citation

  • Peters, J. M., C. J. Nowotarski, and G. L. Mullendore, 2020: Are supercells resistant to entrainment because of their rotation? J. Atmos. Sci., 77, 1475–1495, https://doi.org/10.1175/JAS-D-19-0316.1.

    • Search Google Scholar
    • Export Citation

  • Raymond, D. J., and S. L. Sessions, 2007: Evolution of convection during tropical cyclogenesis. Geophys. Res. Lett., 34, L06811, https://doi.org/10.1029/2006GL028607.

    • Search Google Scholar
    • Export Citation

  • Raymond, D. J., S. L. Sessions, and Ž. Fuchs, 2007: A theory for the spinup of tropical depressions. Quart. J. Roy. Meteor. Soc., 133, 1743–1754, https://doi.org/10.1002/qj.125.

    • Search Google Scholar
    • Export Citation

  • Raymond, D. J., S. L. Sessions, A. H. Sobel, and Ž. Fuchs, 2009: The mechanics of gross moist stability. J. Adv. Model. Earth Syst., 1 (3), https://doi.org/10.3894/JAMES.2009.1.9.

    • Search Google Scholar
    • Export Citation

  • Raymond, D. J., S. L. Sessions, and C. López Carrillo, 2011: Thermodynamics of tropical cyclogenesis in the northwest Pacific. J. Geophys. Res., 116, D18101, https://doi.org/10.1029/2011JD015624.

    • Search Google Scholar
    • Export Citation

  • Raymond, D. J., S. Gjorgjievska, S. Sessions, and Ž. Fuchs, 2014: Tropical cyclogenesis and mid-level vorticity. Aust. Meteor. Oceanogr. J., 64, 11–25.

    • Search Google Scholar
    • Export Citation

  • Romps, D. M., 2014: Rayleigh damping in the free troposphere. J. Atmos. Sci., 71, 553–565, https://doi.org/10.1175/JAS-D-13-062.1.

  • Rotunno, R., and K. A. Emanuel, 1987: An air–sea interaction theory for tropical cyclones. Part II: Evolutionary study using a nonhydrostatic axisymmetric numerical model. J. Atmos. Sci., 44, 542–561, https://doi.org/10.1175/1520-0469(1987)044<0542:AAITFT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Rozoff, C. M., W. H. Schubert, B. D. McNoldy, and J. P. Kossin, 2006: Rapid filamentation zones in intense tropical cyclones. J. Atmos. Sci., 63, 325–340, https://doi.org/10.1175/JAS3595.1.

    • Search Google Scholar
    • Export Citation

  • Schneider, E. K., and R. S. Lindzen, 1976: A discussion of the parameterization of momentum exchange by cumulus convection. J. Geophys. Res., 81, 3158–3160, https://doi.org/10.1029/JC081i018p03158.

    • Search Google Scholar
    • Export Citation

  • Schubert, W. H., and J. J. Hack, 1983: Transformed Eliassen balanced vortex model. J. Atmos. Sci., 40, 1571–1583, https://doi.org/10.1175/1520-0469(1983)040<1571:TEBVM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Schubert, W. H., and B. T. Alworth, 1987: Evolution of potential vorticity in tropical cyclones. Quart. J. Roy. Meteor. Soc., 113, 147–162, https://doi.org/10.1002/qj.49711347509.

    • Search Google Scholar
    • Export Citation

  • Smith, R. K., 2000: The role of cumulus convection in hurricanes and its representation in hurricane models. Rev. Geophys., 38, 465–489, https://doi.org/10.1029/1999RG000080.

    • Search Google Scholar
    • Export Citation

  • Smith, R. K., and S. Vogl, 2008: A simple model of the hurricane boundary layer revisited. Quart. J. Roy. Meteor. Soc., 134, 337–351, https://doi.org/10.1002/qj.216.

    • Search Google Scholar
    • Export Citation

  • Stellmach, S., M. Lischper, K. Julien, G. Vasil, J. S. Cheng, A. Ribeiro, E. M. King, and J. M. Aurnou, 2014: Approaching the asymptotic regime of rapidly rotating convection: Boundary layers versus interior dynamics. Phys. Rev. Lett., 113, 254501, https://doi.org/10.1103/PhysRevLett.113.254501.

    • Search Google Scholar
    • Export Citation

  • Thorpe, A. J., 1985: Diagnosis of balanced vortex structure using potential vorticity. J. Atmos. Sci., 42, 397–406, https://doi.org/10.1175/1520-0469(1985)042<0397:DOBVSU>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Tory, K. J., and M. T. Montgomery, 2008: Tropical cyclone formation: A synopsis of the system-scale development. 28th Conf. on Hurricanes and Tropical Meteorology, Vol. 10, Orlando, FL, Amer. Meteor. Soc., 10A.1, https://ams.confex.com/ams/28Hurricanes/techprogram/paper_138062.htm.

  • Tory, K. J., and W. M. Frank, 2010: Tropical cyclone formation. Global Perspectives on Tropical Cyclones: From Science to Mitigation, J. D. Kepert and J. C. L. Chan, Eds., World Scientific, 55–91.

  • Tory, K. J., M. T. Montgomery, and N. E. Davidson, 2006a: Prediction and diagnosis of tropical cyclone formation in an NWP system. Part I: The critical role of vortex enhancement in deep convection. J. Atmos. Sci., 63, 3077–3090, https://doi.org/10.1175/JAS3764.1.

    • Search Google Scholar
    • Export Citation

  • Tory, K. J., M. T. Montgomery, N. E. Davidson, and J. D. Kepert, 2006b: Prediction and diagnosis of tropical cyclone formation in an NWP system. Part II: A diagnosis of tropical cyclone Chris formation. J. Atmos. Sci., 63, 3091–3113, https://doi.org/10.1175/JAS3765.1.

    • Search Google Scholar
    • Export Citation

  • Vorobieff, P., and R. E. Ecke, 2002: Turbulent rotating convection: An experimental study. J. Fluid Mech., 458, 191–218, https://doi.org/10.1017/S0022112002007814.

    • Search Google Scholar
    • Export Citation

  • Wang, Z., 2014: Role of cumulus congestus in tropical cyclone formation in a high-resolution numerical model simulation. J. Atmos. Sci., 71, 1681–1700, https://doi.org/10.1175/JAS-D-13-0257.1.

    • Search Google Scholar
    • Export Citation

  • Wu, X., and M. Yanai, 1994: Effects of vertical wind shear on the cumulus transport of momentum: Observations and parameterization. J. Atmos. Sci., 51, 1640–1660, https://doi.org/10.1175/1520-0469(1994)051<1640:EOVWSO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Wu, Z., 2003: A shallow CISK, deep equilibrium mechanism for the interaction between large-scale convection and large-scale circulations in the tropics. J. Atmos. Sci., 60, 377–392, https://doi.org/10.1175/1520-0469(2003)060<0377:ASCDEM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Zhang, F., and K. Emanuel, 2016: On the role of surface fluxes and WISHE in tropical cyclone intensification. J. Atmos. Sci., 73, 2011–2019, https://doi.org/10.1175/JAS-D-16-0011.1.

    • Search Google Scholar
    • Export Citation

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  • Arakawa, A., and W. H. Schubert, 1974: Interaction of a cumulus cloud ensemble with the large-scale environment, Part I. J. Atmos. Sci., 31, 674–701, https://doi.org/10.1175/1520-0469(1974)031<0674:IOACCE>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Bell, M. M., and M. T. Montgomery, 2019: Mesoscale processes during the genesis of Hurricane Karl (2010). J. Atmos. Sci., 76, 2235–2255, https://doi.org/10.1175/JAS-D-18-0161.1.

    • Search Google Scholar
    • Export Citation

  • Bister, M., and K. A. Emanuel, 1997: The genesis of Hurricane Guillermo: TEXMEX analyses and a modeling study. Mon. Wea. Rev., 125, 2662–2682, https://doi.org/10.1175/1520-0493(1997)125<2662:TGOHGT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Boos, W. R., and K. A. Emanuel, 2008: Wind–evaporation feedback and abrupt seasonal transitions of weak, axisymmetric Hadley circulations. J. Atmos. Sci., 65, 2194–2214, https://doi.org/10.1175/2007JAS608.1.

    • Search Google Scholar
    • Export Citation

  • Burns, S. P., A. H. Sobel, and L. M. Polvani, 2006: Asymptotic solutions of the axisymmetric moist Hadley circulation in a model with two vertical modes. Theor. Comput. Fluid Dyn., 20, 443–467, https://doi.org/10.1007/s00162-006-0024-z.

    • Search Google Scholar
    • Export Citation

  • Carton, X. J., 1992: On the merger of shielded vortices. Europhys. Lett., 18, 697, https://doi.org/10.1209/0295-5075/18/8/006.

  • Charney, J. G., and A. Eliassen, 1964: On the growth of the hurricane depression. J. Atmos. Sci., 21, 68–75, https://doi.org/10.1175/1520-0469(1964)021<0068:OTGOTH>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Chen, S. S., and W. M. Frank, 1993: A numerical study of the genesis of extratropical convective mesovortices. Part I: Evolution and dynamics. J. Atmos. Sci., 50, 2401–2426, https://doi.org/10.1175/1520-0469(1993)050<2401:ANSOTG>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Clayson, C. A., and L. Kantha, 2008: On turbulence and mixing in the free atmosphere inferred from high-resolution soundings. J. Atmos. Oceanic Technol., 25, 833–852, https://doi.org/10.1175/2007JTECHA992.1.

    • Search Google Scholar
    • Export Citation

  • Davis, C. A., 2015: The formation of moist vortices and tropical cyclones in idealized simulations. J. Atmos. Sci., 72, 3499–3516, https://doi.org/10.1175/JAS-D-15-0027.1.

    • Search Google Scholar
    • Export Citation

  • de Rooy, W. C., and Coauthors, 2013: Entrainment and detrainment in cumulus convection: An overview. Quart. J. Roy. Meteor. Soc., 139 (670), 1–19, https://doi.org/10.1002/qj.1959.

    • Search Google Scholar
    • Export Citation

  • Emanuel, K., 2012: Self-stratification of tropical cyclone outflow. Part II: Implications for storm intensification. J. Atmos. Sci., 69, 988–996, https://doi.org/10.1175/JAS-D-11-0177.1.

    • Search Google Scholar
    • Export Citation

  • Emanuel, K. A., 1986: An air-sea interaction theory for tropical cyclones. Part I: Steady-state maintenance. J. Atmos. Sci., 43, 585–605, https://doi.org/10.1175/1520-0469(1986)043<0585:AASITF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Evans, J. H., 1972: Dimensional analysis and the Buckingham Pi theorem. Amer. J. Phys., 40, 1815–1822, https://doi.org/10.1119/1.1987069.

    • Search Google Scholar
    • Export Citation

  • Fairall, C. W., E. F. Bradley, J. Hare, A. A. Grachev, and J. B. Edson, 2003: Bulk parameterization of air–sea fluxes: Updates and verification for the COARE algorithm. J. Climate, 16, 571–591, https://doi.org/10.1175/1520-0442(2003)016<0571:BPOASF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Fels, S. B., 1982: A parameterization of scale-dependent radiative damping rates in the middle atmosphere. J. Atmos. Sci., 39, 1141–1152, https://doi.org/10.1175/1520-0469(1982)039<1141:APOSDR>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Ferrier, B. S., and R. A. Houze Jr., 1989: One-dimensional time-dependent modeling of GATE cumulonimbus convection. J. Atmos. Sci., 46, 330–352, https://doi.org/10.1175/1520-0469(1989)046<0330:ODTDMO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Fritsch, J. M., J. D. Murphy, and J. S. Kain, 1994: Warm core vortex amplification over land. J. Atmos. Sci., 51, 1780–1807, https://doi.org/10.1175/1520-0469(1994)051<1780:WCVAOL>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Fu, H., and Y. Lin, 2019: A kinematic model for understanding rain formation efficiency of a convective cell. J. Adv. Model. Earth Syst., 11, 4395–4422, https://doi.org/10.1029/2019MS001707.

    • Search Google Scholar
    • Export Citation

  • Fu, H., and M. O’Neill, 2021: The role of random vorticity stretching in tropical depression genesis. J. Atmos. Sci., 78, 4143–4168, https://doi.org/10.1175/JAS-D-21-0087.1.

    • Search Google Scholar
    • Export Citation

  • Fu, H., and S. Sun, 2024: Vorticity skewness of finite-amplitude rapidly rotating Rayleigh–Bénard convection. J. Fluid Mech., 997, A3, https://doi.org/10.1017/jfm.2024.571.

    • Search Google Scholar
    • Export Citation

  • Gjorgjievska, S., and D. J. Raymond, 2014: Interaction between dynamics and thermodynamics during tropical cyclogenesis. Atmos. Chem. Phys., 14, 3065–3082, https://doi.org/10.5194/acp-14-3065-2014.

    • Search Google Scholar
    • Export Citation

  • Hack, J. J., and W. H. Schubert, 1986: Nonlinear response of atmospheric vortices to heating by organized cumulus convection. J. Atmos. Sci., 43, 1559–1573, https://doi.org/10.1175/1520-0469(1986)043<1559:NROAVT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Han, J., and H.-L. Pan, 2006: Sensitivity of hurricane intensity forecast to convective momentum transport parameterization. Mon. Wea. Rev., 134, 664–674, https://doi.org/10.1175/MWR3090.1.

    • Search Google Scholar
    • Export Citation

  • Haynes, P. H., and M. E. McIntyre, 1987: On the evolution of vorticity and potential vorticity in the presence of diabatic heating and frictional or other forces. J. Atmos. Sci., 44, 828–841, https://doi.org/10.1175/1520-0469(1987)044<0828:OTEOVA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Holton, J. R., 2004: An Introduction to Dynamic Meteorology. International Geophysics Series, Vol. 88, Elsevier Science, 535 pp.

  • Houze, R. A., Jr., 1982: Cloud clusters and large-scale vertical motions in the tropics. J. Meteor. Soc. Japan, 60, 396–410, https://doi.org/10.2151/jmsj1965.60.1_396.

    • Search Google Scholar
    • Export Citation

  • Houze, R. A., Jr., 2004: Mesoscale convective systems. Rev. Geophys., 42, RG4003, https://doi.org/10.1029/2004RG000150.

  • Houze, R. A., Jr., W.-C. Lee, and M. M. Bell, 2009: Convective contribution to the genesis of Hurricane Ophelia (2005). Mon. Wea. Rev., 137, 2778–2800, https://doi.org/10.1175/2009MWR2727.1.

    • Search Google Scholar
    • Export Citation

  • Johnson, R. H., T. M. Rickenbach, S. A. Rutledge, P. E. Ciesielski, and W. H. Schubert, 1999: Trimodal characteristics of tropical convection. J. Climate, 12, 2397–2418, https://doi.org/10.1175/1520-0442(1999)012<2397:TCOTC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Julien, K., S. Legg, J. McWilliams, and J. Werne, 1996: Hard turbulence in rotating rayleigh-bénard convection. Phys. Rev. E, 53, R5557, https://doi.org/10.1103/PhysRevE.53.R5557.

    • Search Google Scholar
    • Export Citation

  • Kepert, J., and Y. Wang, 2001: The dynamics of boundary layer jets within the tropical cyclone core. Part II: Nonlinear enhancement. J. Atmos. Sci., 58, 2485–2501, https://doi.org/10.1175/1520-0469(2001)058<2485:TDOBLJ>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Kilroy, G., and R. K. Smith, 2017: The effects of initial vortex size on tropical cyclogenesis and intensification. Quart. J. Roy. Meteor. Soc., 143, 2832–2845, https://doi.org/10.1002/qj.3134.

    • Search Google Scholar
    • Export Citation

  • Kilroy, G., R. K. Smith, and U. Wissmeier, 2014: Tropical convection: The effects of ambient vertical and horizontal vorticity. Quart. J. Roy. Meteor. Soc., 140, 1756–1770, https://doi.org/10.1002/qj.2261.

    • Search Google Scholar
    • Export Citation

  • Kilroy, G., M. T. Montgomery, and R. K. Smith, 2017: The role of boundary-layer friction on tropical cyclogenesis and subsequent intensification. Quart. J. Roy. Meteor. Soc., 143, 2524–2536, https://doi.org/10.1002/qj.3104.

    • Search Google Scholar
    • Export Citation

  • Lebedev, V. I., 1991: On formulae for roots of cubic equation. Russ. J. Numer. Anal. Math. Modell., 6, 315–324, https://doi.org/10.1515/rnam.1991.6.4.315.

    • Search Google Scholar
    • Export Citation

  • Lee, C.-S., 1984: The bulk effects of cumulus momentum transports in tropical cyclones. J. Atmos. Sci., 41, 590–603, https://doi.org/10.1175/1520-0469(1984)041<0590:TBEOCM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • LeMone, M. A., G. M. Barnes, and E. J. Zipser, 1984: Momentum flux by lines of cumulonimbus over the tropical oceans. J. Atmos. Sci., 41, 1914–1932, https://doi.org/10.1175/1520-0469(1984)041<1914:MFBLOC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Liu, C., and M. W. Moncrieff, 2004: Effects of convectively generated gravity waves and rotation on the organization of convection. J. Atmos. Sci., 61, 2218–2227, https://doi.org/10.1175/1520-0469(2004)061<2218:EOCGGW>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Mapes, B. E., 1993: Gregarious tropical convection. J. Atmos. Sci., 50, 2026–2037, https://doi.org/10.1175/1520-0469(1993)050<2026:GTC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Mapes, B. E., 2000: Convective inhibition, subgrid-scale triggering energy, and stratiform instability in a toy tropical wave model. J. Atmos. Sci., 57, 1515–1535, https://doi.org/10.1175/1520-0469(2000)057<1515:CISSTE>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • McWilliams, J. C., and G. R. Flierl, 1979: On the evolution of isolated, nonlinear vortices. J. Phys. Oceanogr., 9, 1155–1182, https://doi.org/10.1175/1520-0485(1979)009<1155:OTEOIN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Moncrieff, M. W., 1992: Organized convective systems: Archetypal dynamical models, mass and momentum flux theory, and parametrization. Quart. J. Roy. Meteor. Soc., 118, 819–850, https://doi.org/10.1002/qj.49711850703.

    • Search Google Scholar
    • Export Citation

  • Montgomery, M. T., and J. Enagonio, 1998: Tropical cyclogenesis via convectively forced vortex Rossby waves in a three-dimensional quasigeostrophic model. J. Atmos. Sci., 55, 3176–3207, https://doi.org/10.1175/1520-0469(1998)055<3176:TCVCFV>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Montgomery, M. T., and R. K. Smith, 2014: Paradigms for tropical cyclone intensification. Aust. Meteor. Oceanogr. J., 64, 37–66.

  • Montgomery, M. T., M. E. Nicholls, T. A. Cram, and A. B. Saunders, 2006: A vortical hot tower route to tropical cyclogenesis. J. Atmos. Sci., 63, 355–386, https://doi.org/10.1175/JAS3604.1.

    • Search Google Scholar
    • Export Citation

  • Montgomery, M. T., G. Kilroy, R. K. Smith, and N. Črnivec, 2020: Contribution of mean and eddy momentum processes to tropical cyclone intensification. Quart. J. Roy. Meteor. Soc., 146, 3101–3117, https://doi.org/10.1002/qj.3837.

    • Search Google Scholar
    • Export Citation

  • Morton, B. R., G. I. Taylor, and J. S. Turner, 1956: Turbulent gravitational convection from maintained and instantaneous sources. Proc. Roy. Soc. London, 234A (1196), 1–23, https://doi.org/10.1098/rspa.1956.0011.

    • Search Google Scholar
    • Export Citation

  • Murthy, V. S., and W. R. Boos, 2019: Understanding the vertical structure of potential vorticity in tropical depressions. Quart. J. Roy. Meteor. Soc., 145, 1968–1991, https://doi.org/10.1002/qj.3539.

    • Search Google Scholar
    • Export Citation

  • Neelin, J. D., and N. Zeng, 2000: A quasi-equilibrium tropical circulation model—Formulation. J. Atmos. Sci., 57, 1741–1766, https://doi.org/10.1175/1520-0469(2000)057<1741:AQETCM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Ogura, Y., and T. Takahashi, 1971: Numerical simulation of the life cycle of a thunderstorm cell. Mon. Wea. Rev., 99, 895–911, https://doi.org/10.1175/1520-0493(1971)099<0895:NSOTLC>2.3.CO;2.

    • Search Google Scholar
    • Export Citation

  • O’Neill, M. E., K. A. Emanuel, and G. R. Flierl, 2016: Weak jets and strong cyclones: Shallow-water modeling of giant planet polar caps. J. Atmos. Sci., 73, 1841–1855, https://doi.org/10.1175/JAS-D-15-0314.1.

    • Search Google Scholar
    • Export Citation

  • Ooyama, K., 1969: Numerical simulation of the life cycle of tropical cyclones. J. Atmos. Sci., 26, 3–40, https://doi.org/10.1175/1520-0469(1969)026<0003:NSOTLC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Persing, J., M. T. Montgomery, J. C. McWilliams, and R. K. Smith, 2013: Asymmetric and axisymmetric dynamics of tropical cyclones. Atmos. Chem. Phys., 13, 12 299–12 341, https://doi.org/10.5194/acp-13-12299-2013.

    • Search Google Scholar
    • Export Citation

  • Peters, J. M., C. J. Nowotarski, and G. L. Mullendore, 2020: Are supercells resistant to entrainment because of their rotation? J. Atmos. Sci., 77, 1475–1495, https://doi.org/10.1175/JAS-D-19-0316.1.

    • Search Google Scholar
    • Export Citation

  • Raymond, D. J., and S. L. Sessions, 2007: Evolution of convection during tropical cyclogenesis. Geophys. Res. Lett., 34, L06811, https://doi.org/10.1029/2006GL028607.

    • Search Google Scholar
    • Export Citation

  • Raymond, D. J., S. L. Sessions, and Ž. Fuchs, 2007: A theory for the spinup of tropical depressions. Quart. J. Roy. Meteor. Soc., 133, 1743–1754, https://doi.org/10.1002/qj.125.

    • Search Google Scholar
    • Export Citation

  • Raymond, D. J., S. L. Sessions, A. H. Sobel, and Ž. Fuchs, 2009: The mechanics of gross moist stability. J. Adv. Model. Earth Syst., 1 (3), https://doi.org/10.3894/JAMES.2009.1.9.

    • Search Google Scholar
    • Export Citation

  • Raymond, D. J., S. L. Sessions, and C. López Carrillo, 2011: Thermodynamics of tropical cyclogenesis in the northwest Pacific. J. Geophys. Res., 116, D18101, https://doi.org/10.1029/2011JD015624.

    • Search Google Scholar
    • Export Citation

  • Raymond, D. J., S. Gjorgjievska, S. Sessions, and Ž. Fuchs, 2014: Tropical cyclogenesis and mid-level vorticity. Aust. Meteor. Oceanogr. J., 64, 11–25.

    • Search Google Scholar
    • Export Citation

  • Romps, D. M., 2014: Rayleigh damping in the free troposphere. J. Atmos. Sci., 71, 553–565, https://doi.org/10.1175/JAS-D-13-062.1.

  • Rotunno, R., and K. A. Emanuel, 1987: An air–sea interaction theory for tropical cyclones. Part II: Evolutionary study using a nonhydrostatic axisymmetric numerical model. J. Atmos. Sci., 44, 542–561, https://doi.org/10.1175/1520-0469(1987)044<0542:AAITFT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Rozoff, C. M., W. H. Schubert, B. D. McNoldy, and J. P. Kossin, 2006: Rapid filamentation zones in intense tropical cyclones. J. Atmos. Sci., 63, 325–340, https://doi.org/10.1175/JAS3595.1.

    • Search Google Scholar
    • Export Citation

  • Schneider, E. K., and R. S. Lindzen, 1976: A discussion of the parameterization of momentum exchange by cumulus convection. J. Geophys. Res., 81, 3158–3160, https://doi.org/10.1029/JC081i018p03158.

    • Search Google Scholar
    • Export Citation

  • Schubert, W. H., and J. J. Hack, 1983: Transformed Eliassen balanced vortex model. J. Atmos. Sci., 40, 1571–1583, https://doi.org/10.1175/1520-0469(1983)040<1571:TEBVM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Schubert, W. H., and B. T. Alworth, 1987: Evolution of potential vorticity in tropical cyclones. Quart. J. Roy. Meteor. Soc., 113, 147–162, https://doi.org/10.1002/qj.49711347509.

    • Search Google Scholar
    • Export Citation

  • Smith, R. K., 2000: The role of cumulus convection in hurricanes and its representation in hurricane models. Rev. Geophys., 38, 465–489, https://doi.org/10.1029/1999RG000080.

    • Search Google Scholar
    • Export Citation

  • Smith, R. K., and S. Vogl, 2008: A simple model of the hurricane boundary layer revisited. Quart. J. Roy. Meteor. Soc., 134, 337–351, https://doi.org/10.1002/qj.216.

    • Search Google Scholar
    • Export Citation

  • Stellmach, S., M. Lischper, K. Julien, G. Vasil, J. S. Cheng, A. Ribeiro, E. M. King, and J. M. Aurnou, 2014: Approaching the asymptotic regime of rapidly rotating convection: Boundary layers versus interior dynamics. Phys. Rev. Lett., 113, 254501, https://doi.org/10.1103/PhysRevLett.113.254501.

    • Search Google Scholar
    • Export Citation

  • Thorpe, A. J., 1985: Diagnosis of balanced vortex structure using potential vorticity. J. Atmos. Sci., 42, 397–406, https://doi.org/10.1175/1520-0469(1985)042<0397:DOBVSU>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Tory, K. J., and M. T. Montgomery, 2008: Tropical cyclone formation: A synopsis of the system-scale development. 28th Conf. on Hurricanes and Tropical Meteorology, Vol. 10, Orlando, FL, Amer. Meteor. Soc., 10A.1, https://ams.confex.com/ams/28Hurricanes/techprogram/paper_138062.htm.

  • Tory, K. J., and W. M. Frank, 2010: Tropical cyclone formation. Global Perspectives on Tropical Cyclones: From Science to Mitigation, J. D. Kepert and J. C. L. Chan, Eds., World Scientific, 55–91.

  • Tory, K. J., M. T. Montgomery, and N. E. Davidson, 2006a: Prediction and diagnosis of tropical cyclone formation in an NWP system. Part I: The critical role of vortex enhancement in deep convection. J. Atmos. Sci., 63, 3077–3090, https://doi.org/10.1175/JAS3764.1.

    • Search Google Scholar
    • Export Citation

  • Tory, K. J., M. T. Montgomery, N. E. Davidson, and J. D. Kepert, 2006b: Prediction and diagnosis of tropical cyclone formation in an NWP system. Part II: A diagnosis of tropical cyclone Chris formation. J. Atmos. Sci., 63, 3091–3113, https://doi.org/10.1175/JAS3765.1.

    • Search Google Scholar
    • Export Citation

  • Vorobieff, P., and R. E. Ecke, 2002: Turbulent rotating convection: An experimental study. J. Fluid Mech., 458, 191–218, https://doi.org/10.1017/S0022112002007814.

    • Search Google Scholar
    • Export Citation

  • Wang, Z., 2014: Role of cumulus congestus in tropical cyclone formation in a high-resolution numerical model simulation. J. Atmos. Sci., 71, 1681–1700, https://doi.org/10.1175/JAS-D-13-0257.1.

    • Search Google Scholar
    • Export Citation

  • Wu, X., and M. Yanai, 1994: Effects of vertical wind shear on the cumulus transport of momentum: Observations and parameterization. J. Atmos. Sci., 51, 1640–1660, https://doi.org/10.1175/1520-0469(1994)051<1640:EOVWSO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Wu, Z., 2003: A shallow CISK, deep equilibrium mechanism for the interaction between large-scale convection and large-scale circulations in the tropics. J. Atmos. Sci., 60, 377–392, https://doi.org/10.1175/1520-0469(2003)060<0377:ASCDEM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Zhang, F., and K. Emanuel, 2016: On the role of surface fluxes and WISHE in tropical cyclone intensification. J. Atmos. Sci., 73, 2011–2019, https://doi.org/10.1175/JAS-D-16-0011.1.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    A schematic of the single-column model that depicts the horizontally averaged quantities in a narrow cylinder that covers a portion of the vortex’s inner core. The boundary layer top is defined as the origin of the vertical coordinate, with z = −HB depicting the sea surface and z = H depicting the tropopause. The blue-shaded region denotes the boundary layer, where the surface friction induces an Ekman circulation (black arrows) that decays with height. The red arrows denote cyclonic convergent flows, and the blue arrows denote anticyclonic divergent flows.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    A schematic diagram of the vertical mode decomposition of vertical velocity and vertical vorticity, without considering surface friction. The first row shows the vertical velocity of the n = 0 mode (vertically uniform part), n = 1 mode [sin(πz/H)], n = 2 mode [sin(2πz/H)], and their sum: w = ΔW sin (π z/H) − AΔW sin (2πz/H), plotted with solid blue lines. The solid black lines denote the lower and upper boundaries. The dashed red line denotes the zero reference line. The second row shows the vertical vorticity. The Z0, Z1, and Z2 are the amplitude of the n = 0 mode, n = 1 mode, and n = 2 mode. Their magnitudes are sketched and not from any rigorous solution. A typical TC precursor vortex has Z0 > 0, Z1 > 0, and Z2 < 0.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    The time evolution of the vertical structure of ζz/f for the three numerical experiments with the column model, all using A = 0.5. The increment of the contour line is 0.25. The zero contour line is thickened. (a) The τ*/τd=0.25 experiment. (b) The τ*/τd=1 experiment. (c) The τ*/τd=2 experiment.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    (a) Comparing the surface vorticity (normalized by f) predicted by the column model (solid lines) and the analytical solution of the reduced model (dashed lines). The blue, red, and yellow lines denote τ*/τd=0.25, 1, and 2 experiments. All of them use A = 0.5. (b) As in (a), but the dashed lines are replaced by dotted lines, representing the column model’s surface vorticity output subtracting the contribution of the barotropic vorticity: (ζz,surf−ζz¯)/f.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    The dependence of surface vorticity (normalized by f) on τ*/τd at t/τ*=2 time. (a) The A = 0.25 experiments. (b) The A = 0.5 experiments. (c) The A = 1 experiments. The solid lines denote the results of the column model, the dotted lines denote the results of the column model subtracting the barotropic vorticity [(ζz,surf−ζz¯)/f], and the dashed lines denote the analytical solution of the reduced model. The difference between the solid and dotted lines denotes the contribution from the barotropic vorticity.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    A schematic diagram of (first row) how deep convective heating and (second row) stratiform heating produce barotropic vorticity via (first column) the stretching of vertical vorticity and (second column) the vertical advection of vertical vorticity. (third column) The sum of the two terms. Only the vorticity tendency produced by the same mode index is considered. The first row uses the n = 1 type of w and ζz; the second row uses the n = 2 type of w and ζz.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    The time evolution of the vorticity amplitude Zn for n = 0 (blue line), n = 1 (red line), n = 2 (yellow line), and n = 3 (purple line) vertical modes. The solid lines denote the column model’s results, which are obtained from the Fourier transform of the solution. The dashed lines denote the reduced equation’s results. (a) The τ*/τd=0.25 experiment. (b) The τ*/τd=1 experiment. (c) The τ*/τd=2 experiment. All experiments use A = 0.5.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    (a) The analytical solution of ζz,surf/f at t/τ*=2 time with varying τ*/τd and A. The contour interval is 0.2. A higher τ*/τd means stronger diffusion (cumulus friction), and a higher A means a more top-heavy diabatic heating profile. The white crosses denote the three experiments with A = 0.5 and τ*/τd=0.25, 1, and 2. (b) As in (a), but letting λ2 = λ3 = 0 in Λ [Eq. (41)] to isolate the effect of the largest eigenvalue λ1. In (b), the regime where one of the eigenvalues is imaginary is denoted as the white region. This is a weakly diffusive regime where the truncation assumption of the reduced equation is invalid and does not deserve further investigation. (second row) As in (first row), but that they show the surface vorticity subtracting the barotropic vorticity, and the (ζz,surf−ζz¯)/f=0 contour line is bold.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    The result of numerical experiments with τ∗/τd=0.5, A = 0.5, and varying surface drag coefficients CD. (a) The time evolution of the surface vorticity, with the time normalized by τ∗ and the vorticity normalized by f. The experiments with CD = 0, 0.001, 0.002, and 0.003 are shown as solid lines whose color gradually turns lighter. (b) The surface vorticity versus the barotropic vorticity. Both quantities are normalized by f. (c) The barotropic vorticity (normalized by f) versus the square of the vortex Rossby number. (d) The surface vorticity (normalized by f) versus the square of the vortex Rossby number. The dashed black lines in (b)–(d) are 1:1 reference lines.

  • <sc>Fig</sc>. 10.
    View in gallery
    Fig. 10.

    The three dotted lines show the critical Ro diagnosed with the dev = 0.1 threshold in three groups of experiments with τ*/τd=0.25 (blue), τ*/τd=0.5 (red), and τ*/τd=1 (yellow). Each line shows 10 experiments with CD uniformly sampled between 0.001 and 0.01. The plotting uses the log–log coordinate. The solid black line marks the CD−1/3 reference slope (its magnitude does not have any specific meaning).

Fig. 1.

A schematic of the single-column model that depicts the horizontally averaged quantities in a narrow cylinder that covers a portion of the vortex’s inner core. The boundary layer top is defined as the origin of the vertical coordinate, with z = −HB depicting the sea surface and z = H depicting the tropopause. The blue-shaded region denotes the boundary layer, where the surface friction induces an Ekman circulation (black arrows) that decays with height. The red arrows denote cyclonic convergent flows, and the blue arrows denote anticyclonic divergent flows.
Fig. 2.

A schematic diagram of the vertical mode decomposition of vertical velocity and vertical vorticity, without considering surface friction. The first row shows the vertical velocity of the n = 0 mode (vertically uniform part), n = 1 mode [sin(πz/H)], n = 2 mode [sin(2πz/H)], and their sum: w = ΔW sin (π z/H) − AΔW sin (2πz/H), plotted with solid blue lines. The solid black lines denote the lower and upper boundaries. The dashed red line denotes the zero reference line. The second row shows the vertical vorticity. The Z0, Z1, and Z2 are the amplitude of the n = 0 mode, n = 1 mode, and n = 2 mode. Their magnitudes are sketched and not from any rigorous solution. A typical TC precursor vortex has Z0 > 0, Z1 > 0, and Z2 < 0.
Fig. 3.

The time evolution of the vertical structure of ζz/f for the three numerical experiments with the column model, all using A = 0.5. The increment of the contour line is 0.25. The zero contour line is thickened. (a) The τ*/τd=0.25 experiment. (b) The τ*/τd=1 experiment. (c) The τ*/τd=2 experiment.
Fig. 4.

(a) Comparing the surface vorticity (normalized by f) predicted by the column model (solid lines) and the analytical solution of the reduced model (dashed lines). The blue, red, and yellow lines denote τ*/τd=0.25, 1, and 2 experiments. All of them use A = 0.5. (b) As in (a), but the dashed lines are replaced by dotted lines, representing the column model’s surface vorticity output subtracting the contribution of the barotropic vorticity: (ζz,surf−ζz¯)/f.
Fig. 5.

The dependence of surface vorticity (normalized by f) on τ*/τd at t/τ*=2 time. (a) The A = 0.25 experiments. (b) The A = 0.5 experiments. (c) The A = 1 experiments. The solid lines denote the results of the column model, the dotted lines denote the results of the column model subtracting the barotropic vorticity [(ζz,surf−ζz¯)/f], and the dashed lines denote the analytical solution of the reduced model. The difference between the solid and dotted lines denotes the contribution from the barotropic vorticity.
Fig. 6.

A schematic diagram of (first row) how deep convective heating and (second row) stratiform heating produce barotropic vorticity via (first column) the stretching of vertical vorticity and (second column) the vertical advection of vertical vorticity. (third column) The sum of the two terms. Only the vorticity tendency produced by the same mode index is considered. The first row uses the n = 1 type of w and ζz; the second row uses the n = 2 type of w and ζz.
Fig. 7.

The time evolution of the vorticity amplitude Zn for n = 0 (blue line), n = 1 (red line), n = 2 (yellow line), and n = 3 (purple line) vertical modes. The solid lines denote the column model’s results, which are obtained from the Fourier transform of the solution. The dashed lines denote the reduced equation’s results. (a) The τ*/τd=0.25 experiment. (b) The τ*/τd=1 experiment. (c) The τ*/τd=2 experiment. All experiments use A = 0.5.
Fig. 8.

(a) The analytical solution of ζz,surf/f at t/τ*=2 time with varying τ*/τd and A. The contour interval is 0.2. A higher τ*/τd means stronger diffusion (cumulus friction), and a higher A means a more top-heavy diabatic heating profile. The white crosses denote the three experiments with A = 0.5 and τ*/τd=0.25, 1, and 2. (b) As in (a), but letting λ2 = λ3 = 0 in Λ [Eq. (41)] to isolate the effect of the largest eigenvalue λ1. In (b), the regime where one of the eigenvalues is imaginary is denoted as the white region. This is a weakly diffusive regime where the truncation assumption of the reduced equation is invalid and does not deserve further investigation. (second row) As in (first row), but that they show the surface vorticity subtracting the barotropic vorticity, and the (ζz,surf−ζz¯)/f=0 contour line is bold.
Fig. 9.

The result of numerical experiments with τ∗/τd=0.5, A = 0.5, and varying surface drag coefficients CD. (a) The time evolution of the surface vorticity, with the time normalized by τ∗ and the vorticity normalized by f. The experiments with CD = 0, 0.001, 0.002, and 0.003 are shown as solid lines whose color gradually turns lighter. (b) The surface vorticity versus the barotropic vorticity. Both quantities are normalized by f. (c) The barotropic vorticity (normalized by f) versus the square of the vortex Rossby number. (d) The surface vorticity (normalized by f) versus the square of the vortex Rossby number. The dashed black lines in (b)–(d) are 1:1 reference lines.
Fig. 10.

The three dotted lines show the critical Ro diagnosed with the dev = 0.1 threshold in three groups of experiments with τ*/τd=0.25 (blue), τ*/τd=0.5 (red), and τ*/τd=1 (yellow). Each line shows 10 experiments with CD uniformly sampled between 0.001 and 0.01. The plotting uses the log–log coordinate. The solid black line marks the CD−1/3 reference slope (its magnitude does not have any specific meaning).
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